Number of the day: 2477

Carl Ludwig Siegel was born on this day 122 years ago.

Properties of the number 2477:

2477 is a cyclic number.
2477 is the 367^th prime.
2477 has 9 antidivisors and 2476 totatives.
2477 has an oblong digit sum 20 in base 10.
2477 = 4³ + 6³ + 13³ is the sum of 3 positive cubes in 1 way.
2477 = 1239² - 1238² is the difference of 2 nonnegative squares in 1 way.
2477 is the difference of 2 positive pentagonal numbers in 1 way.
2477 = 19² + 46² is the sum of 2 positive squares in 1 way.
2477 = 2² + 13² + 48² is the sum of 3 positive squares.
2477² = 1748² + 1755² is the sum of 2 positive squares in 1 way.
2477² is the sum of 3 positive squares.
2477 is a proper divisor of 19⁶¹⁹ - 1.
2477 is an emirp in (at least) the following bases: 3, 5, 15, 18, 19, 31, 36, 37, 39, 40, 42, 43, 46, 47, 49, 50, 53, 59, 60, 61, 62, 66, 67, 71, 77, 79, 83, 85, 91, 95, 96, 98, and 99.
2477 is palindromic in (at least) the following bases: 33, -14, -19, -24, and -45.
2477 in base 18 = 7bb and consists of only the digits '7' and 'b'.
2477 in base 32 = 2dd and consists of only the digits '2' and 'd'.
2477 in base 33 = 292 and consists of only the digits '2' and '9'.
2477 in base 49 = 11R and consists of only the digits '1' and 'R'.

The number 2477 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000230: Smallest prime p such that there is a gap of 2n between p and next prime.
A038580: Primes with indices that are primes with prime indices.
A049079: Primes prime(k) for which A049076(k) = 3.
A053001: Largest prime < n^2.
A054804: First term of strong prime quartets: prime(m+1)-prime(m) > prime(m+2)-prime(m+1) > prime(m+3)-prime(m+2).
A057190: Numbers n such that (24^n+1)/25 is a prime.
A075580: Smallest prime p(k) such that the number of distinct prime divisors of all composite numbers between p(k) and p(k+1) is n.
A232047: T(n,k)=Number of nXk 0..1 arrays with no element less than a strict majority of its horizontal and antidiagonal neighbors
A272285: Primes of the form 43*n^2 - 537*n + 2971 in order of increasing nonnegative values of n.
A299264: Partial sums of A299258.

Number of the day: 3506

Properties of the number 3506:

3506 = 2 × 1753 is semiprime and squarefree.
3506 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 1752 totatives.
3506 has a semiprime digit sum 14 in base 10.
3506 has sum of divisors equal to 5262 which is a sphenic number.
Reversing the decimal digits of 3506 results in a prime.
3506 = 5² + 59² is the sum of 2 positive squares in 1 way.
3506 = 1² + 16² + 57² is the sum of 3 positive squares.
3506² = 590² + 3456² is the sum of 2 positive squares in 1 way.
3506² is the sum of 3 positive squares.
3506 is a proper divisor of 1571⁶ - 1.
3506 is an emirpimes in (at least) the following bases: 2, 3, 4, 5, 6, 7, 12, 15, 17, 18, 19, 26, 27, 29, 32, 34, 35, 39, 41, 48, 51, 52, 54, 55, 58, 60, 63, 64, 66, 71, 73, 74, 77, 81, 84, 87, 90, 91, 92, 93, 94, 98, and 100.
3506 is palindromic in (at least) the following bases: 31, -5, -19, -22, -25, -34, and -48.
3506 in base 8 = 6662 and consists of only the digits '2' and '6'.
3506 in base 18 = aee and consists of only the digits 'a' and 'e'.
3506 in base 24 = 622 and consists of only the digits '2' and '6'.
3506 in base 29 = 44q and consists of only the digits '4' and 'q'.
3506 in base 30 = 3qq and consists of only the digits '3' and 'q'.
3506 in base 31 = 3k3 and consists of only the digits '3' and 'k'.

The number 3506 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000070: a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).
A025065: Number of palindromic partitions of n.
A025414: a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways.
A071609: Squared radii of the spheres around (0,0,0) that contain record numbers of lattice points.
A077055: Call two meanders from A005316 equivalent if they differ by a reflection in the Y axis (if n even) or by reflections in the X or Y axes (if n odd). Sequence gives number of inequivalent meanders with n crossings.
A119735: Numbers n such that every digit occurs at least once in n^3.
A120919: Cascadence of (1+x)^3; a triangle, read by rows of 3n+1 terms, that retains its original form upon convolving each row with [1,3,3,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 3n+1 terms remain in row n for n>=0.
A154493: a(n+1)-+a(n)=prime, a(n+1)*a(n)=Average of twin prime pairs, a(0)=1,a(1)=4.
A224728: T(n,k)=Number of (n+3)X(k+3) 0..2 matrices with each 4X4 subblock idempotent
A281541: Expansion of Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^(j^2)).

Number of the day: 769169

Properties of the number 769169:

769169 is a cyclic number.
769169 is the 61674^th prime.
769169 has 9 antidivisors and 769168 totatives.
769169 has a semiprime digit sum 38 in base 10.
Reversing the decimal digits of 769169 results in a semiprime.
769169 = 384585² - 384584² is the difference of 2 nonnegative squares in 1 way.
769169 is the difference of 2 positive pentagonal numbers in 1 way.
769169 = 385² + 788² is the sum of 2 positive squares in 1 way.
769169 = 13² + 110² + 870² is the sum of 3 positive squares.
769169² = 472719² + 606760² is the sum of 2 positive squares in 1 way.
769169² is the sum of 3 positive squares.
769169 is a proper divisor of 101⁴⁸⁰⁷³ - 1.
769169 is an emirp in (at least) the following bases: 7, 11, 13, 14, 17, 18, 26, 27, 28, 29, 34, 45, 46, 55, 61, 71, 78, 79, 85, 87, 89, and 97.

Number of the day: 27243

John von Neumann was born on this day 115 years ago.

Properties of the number 27243:

27243 = 3³ × 1009 is the 24260^th composite number and is not squarefree.
27243 has 2 distinct prime factors, 8 divisors, 17 antidivisors and 18144 totatives.
27243 = 3³ + 6³ + 30³ = 12³ + 18³ + 27³ is the sum of 3 positive cubes in 2 ways.
27243 = 13622² - 13621² = 4542² - 4539² = 1518² - 1509² = 518² - 491² is the difference of 2 nonnegative squares in 4 ways.
27243 is the difference of 2 positive pentagonal numbers in 1 way.
27243 = 7² + 25² + 163² is the sum of 3 positive squares.
27243² = 15093² + 22680² is the sum of 2 positive squares in 1 way.
27243² is the sum of 3 positive squares.
27243 is a proper divisor of 337⁹ - 1.
27243 = '2' + '7243' is the concatenation of 2 prime numbers.
27243 in base 3 = 1101101000 and consists of only the digits '0' and '1'.

The number 27243 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002255: Numbers n such that 7*4^n + 1 is prime.
A009474: a(n) is the concatenation of n and 9n.
A191573: Number of n X n symmetric binary matrices with each 1 adjacent to no more than 5 knight-move neighboring 1s
A225967: Number of permutations of [n] having exactly 6 strong fixed blocks.
A228064: Difference between the number of primes with n digits (A006879) and the nearest integer to F[4n](S(n)), where F[4n](x) are Fibonacci polynomials and S(n) = Sum_{i=0..3} (C(i)*(log(log(A*(B+n^2))))^i) (see A228063).

Number of the day: 980312690405418

Johannes Kepler was born on this day 447 years ago.

Johannes Kepler was born on this day 447 years ago.

Jacob Bernoulli was born on this day 364 years ago.

Properties of the number 980312690405418:

980312690405418 = 2 × 3 × 31 × 53 × 1123 × 4723 × 18749 is composite and squarefree.
980312690405418 has 7 distinct prime factors, 128 divisors, 75 antidivisors and 309904853955840 totatives.
980312690405418 = 59357³ + 71480³ + 74045³ is the sum of 3 positive cubes in 1 way.
980312690405418 is the sum of 2 positive triangular numbers.
980312690405418 is the difference of 2 positive pentagonal numbers in 15 ways.
980312690405418 = 220² + 12637² + 31309943² is the sum of 3 positive squares.
980312690405418² = 517901043987768² + 832340963551770² = 359348774368470² + 912075341915232² = 80571943007568² + 976995973875990² = 584558956676760² + 786958573966182² is the sum of 2 positive squares in 4 ways.
980312690405418² is the sum of 3 positive squares.
980312690405418 is a proper divisor of 433^22958364 - 1.
980312690405418 = '98031' + '2690405418' is the concatenation of 2 sphenic numbers.

Number of the day: 93538

Charles Babbage was born on this day 227 years ago.

Properties of the number 93538:

93538 = 2 × 46769 is semiprime and squarefree.
93538 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 46768 totatives.
93538 has a triangular digit sum 28 in base 10.
93538 has a triangular digit product 3240 in base 10.
Reversing the decimal digits of 93538 results in an emirpimes.
93538 = 6³ + 13³ + 45³ is the sum of 3 positive cubes in 1 way.
93538 is the sum of 2 positive triangular numbers.
93538 is the difference of 2 positive pentagonal numbers in 2 ways.
93538 = 73² + 297² is the sum of 2 positive squares in 1 way.
93538 = 28² + 135² + 273² is the sum of 3 positive squares.
93538² = 43362² + 82880² is the sum of 2 positive squares in 1 way.
93538² is the sum of 3 positive squares.
93538 is a proper divisor of 1213¹⁴⁸ - 1.
93538 = '93' + '538' is the concatenation of 2 emirpimes.
93538 is an emirpimes in (at least) the following bases: 5, 8, 10, 11, 19, 20, 26, 27, 30, 31, 32, 34, 35, 37, 38, 39, 44, 45, 49, 50, 51, 59, 60, 63, 64, 69, 70, 74, 75, 80, 85, 86, 90, 92, 96, 98, and 100.
93538 is palindromic in (at least) the following bases: 61, -16, -50, -78, and -87.
93538 in base 40 = 1III and consists of only the digits '1' and 'I'.
93538 in base 49 = ckk and consists of only the digits 'c' and 'k'.
93538 in base 60 = Pww and consists of only the digits 'P' and 'w'.
93538 in base 61 = P8P and consists of only the digits '8' and 'P'.

The number 93538 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A065455: Number of (binary) bit strings of length n in which no even block of 0's is followed by an odd block of 1's.
A109084: G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.
A212830: Number of 0..n arrays of length n+1 with no adjacent pair equal to its immediately preceding adjacent pair, and new values introduced in 0..n order
A296918: List of coefficients of reduced chromatic polynomial of dodecahedron, highest order terms first.

Number of the day: 48739

Properties of the number 48739:

48739 is a cyclic number.
48739 = 17 × 47 × 61 is a sphenic number and squarefree.
48739 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 44160 totatives.
48739 has an emirp digit sum 31 in base 10.
48739 = 24370² - 24369² = 1442² - 1425² = 542² - 495² = 430² - 369² is the difference of 2 nonnegative squares in 4 ways.
48739 is the difference of 2 positive pentagonal numbers in 4 ways.
48739 = 7² + 27² + 219² is the sum of 3 positive squares.
48739² = 22936² + 43005² = 14805² + 46436² = 30315² + 38164² = 8789² + 47940² is the sum of 2 positive squares in 4 ways.
48739² is the sum of 3 positive squares.
48739 is a proper divisor of 1597⁴ - 1.
48739 = '4873' + '9' is the concatenation of 2 semiprime numbers.

The number 48739 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A185265: a(0)=1, a(1)=2; thereafter a(n) = f(n-1) + f(n-2) where f() = A164387().

Number of the day: 449211

Srinivasa Ramanujan was born on this day 131 years ago.

Properties of the number 449211:

449211 = 3 × 7 × 21391 is a sphenic number and squarefree.
449211 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 256680 totatives.
449211 has a semiprime digit sum 21 in base 10.
449211 has a Fibonacci digit sum 21 in base 10.
449211 has a triangular digit sum 21 in base 10.
449211 = 224606² - 224605² = 74870² - 74867² = 32090² - 32083² = 10706² - 10685² is the difference of 2 nonnegative squares in 4 ways.
449211 is the difference of 2 positive pentagonal numbers in 2 ways.
449211 = 29² + 59² + 667² is the sum of 3 positive squares.
449211² is the sum of 3 positive squares.
449211 is a proper divisor of 241¹³⁸ - 1.
449211 = '449' + '211' is the concatenation of 2 prime numbers.
449211 is palindromic in (at least) base 84.

Number of the day: 2870581

Properties of the number 2870581:

2870581 = 7 × 127 × 3229 is a sphenic number and squarefree.
2870581 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 2440368 totatives.
2870581 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 2870581 results in a semiprime.
2870581 = 1435291² - 1435290² = 205045² - 205038² = 11365² - 11238² = 2059² - 1170² is the difference of 2 nonnegative squares in 4 ways.
2870581 is the sum of 2 positive triangular numbers.
2870581 is the difference of 2 positive pentagonal numbers in 4 ways.
2870581 = 34² + 81² + 1692² is the sum of 3 positive squares.
2870581² = 1574419² + 2400300² is the sum of 2 positive squares in 1 way.
2870581² is the sum of 3 positive squares.
2870581 is a proper divisor of 839⁸⁴ - 1.

Number of the day: 13196

Properties of the number 13196:

13196 = 2² × 3299 is the 11625^th composite number and is not squarefree.
13196 has 2 distinct prime factors, 6 divisors, 7 antidivisors and 6596 totatives.
13196 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 13196 results in a semiprime.
13196 = 3300² - 3298² is the difference of 2 nonnegative squares in 1 way.
13196 is the difference of 2 positive pentagonal numbers in 1 way.
13196 = 14² + 30² + 110² is the sum of 3 positive squares.
13196² is the sum of 3 positive squares.
13196 is a proper divisor of 179³⁴ - 1.
13196 is palindromic in (at least) the following bases: 25, 32, 91, -8, -38, -42, and -79.
13196 in base 3 = 200002202 and consists of only the digits '0' and '2'.
13196 in base 12 = 7778 and consists of only the digits '7' and '8'.
13196 in base 25 = l2l and consists of only the digits '2' and 'l'.
13196 in base 32 = csc and consists of only the digits 'c' and 's'.

The number 13196 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A023621: Convolution of Lucas numbers and A000201.
A024532: a(n) = [ (3rd elementary symmetric function of P(n))/(first elementary symmetric function of P(n)) ], where P(n) = {1, p(1), p(2), ..., p(n-1)}, where p(0) = 1.
A055291: Number of trees with n nodes and 4 leaves.
A104909: a(n) = A104908(n) - 10*A104863(n).
A117496: Numbers with no 1's in base 3 & 4 expansions.
A121718: Write 0, 1, ..., n in base 3 and add as if they were decimal numbers.
A224668: Number of 6 X 6 0..n matrices with each 2 X 2 subblock idempotent.
A253503: Number of (n+2) X (1+2) 0..1 arrays with every 2 X 2 and 3 X 3 subblock diagonal maximum minus antidiagonal minimum nondecreasing horizontally and vertically.
A258440: Number of squares of all sizes in polyominoes obtained by union of two bi-symmetric figures (A241526) with intersection equals A173196.
A272453: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 478", based on the 5-celled von Neumann neighborhood.

Number of the day: 59868970

Properties of the number 59868970:

59868970 = 2 × 5 × 7 × 855271 is the 56314141^th composite number and is squarefree.
59868970 has 4 distinct prime factors, 16 divisors, 9 antidivisors and 20526480 totatives.
59868970 is the sum of 2 positive triangular numbers.
59868970 is the difference of 2 positive pentagonal numbers in 4 ways.
59868970 = 103² + 300² + 7731² is the sum of 3 positive squares.
59868970² = 35921382² + 47895176² is the sum of 2 positive squares in 1 way.
59868970² is the sum of 3 positive squares.
59868970 is a proper divisor of 491⁴⁵ - 1.

Number of the day: 67866

Marius Sophus Lie was born on this day 176 years ago.

Properties of the number 67866:

67866 = 2 × 3 × 11311 is a sphenic number and squarefree.
67866 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 22620 totatives.
67866 has a semiprime digit sum 33 in base 10.
67866 is the sum of 2 positive triangular numbers.
67866 = 56² + 61² + 247² is the sum of 3 positive squares.
67866² is the sum of 3 positive squares.
67866 is a proper divisor of 967¹³ - 1.
67866 = '678' + '66' is the concatenation of 2 sphenic numbers.
67866 = '6786' + '6' is the concatenation of 2 triangular numbers.
67866 in base 12 = 33336 and consists of only the digits '3' and '6'.

The number 67866 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A104268: a(n) = 2*4^(n-1) - (3n-1)/(2n+2)*C(2n,n).

Number of the day: 4590770284305

Properties of the number 4590770284305:

4590770284305 = 3² × 5 × 43 × 23549 × 100747 is composite and not squarefree.
4590770284305 has 5 distinct prime factors, 48 divisors, 53 antidivisors and 2391345742464 totatives.
4590770284305 is the difference of 2 nonnegative squares in 24 ways.
4590770284305 is the difference of 2 positive pentagonal numbers in 7 ways.
4590770284305 = 280² + 1079² + 2142608² is the sum of 3 positive squares.
4590770284305² = 2754462170583² + 3672616227444² = 2651881877424² + 3747358311057² = 2854937052936² + 3595066929423² = 126909484695² + 4589015775300² is the sum of 2 positive squares in 4 ways.
4590770284305² is the sum of 3 positive squares.
4590770284305 is a proper divisor of 389²³⁵⁰⁷⁴ - 1.
4590770284305 = '45907702' + '84305' is the concatenation of 2 sphenic numbers.

Number of the day: 4948

János Bolyai was born on this day 216 years ago.

Properties of the number 4948:

4948 = 2² × 1237 is the 4286^th composite number and is not squarefree.
4948 has 2 distinct prime factors, 6 divisors, 5 antidivisors and 2472 totatives.
4948 has a semiprime digit sum 25 in base 10.
4948 has sum of divisors equal to 8666 which is a sphenic number.
Reversing the decimal digits of 4948 results in a sphenic number.
4948 = 2³ + 3³ + 17³ is the sum of 3 positive cubes in 1 way.
4948 = 1238² - 1236² is the difference of 2 nonnegative squares in 1 way.
4948 is the sum of 2 positive triangular numbers.
4948 is the difference of 2 positive pentagonal numbers in 1 way.
4948 = 18² + 68² is the sum of 2 positive squares in 1 way.
4948 = 18² + 32² + 60² is the sum of 3 positive squares.
4948² = 2448² + 4300² is the sum of 2 positive squares in 1 way.
4948² is the sum of 3 positive squares.
4948 is a proper divisor of 691⁴ - 1.
4948 is palindromic in (at least) the following bases: 51, -18, -26, -27, -43, and -97.
4948 in base 19 = dd8 and consists of only the digits '8' and 'd'.
4948 in base 26 = 788 and consists of only the digits '7' and '8'.
4948 in base 40 = 33S and consists of only the digits '3' and 'S'.
4948 in base 49 = 22m and consists of only the digits '2' and 'm'.
4948 in base 50 = 1mm and consists of only the digits '1' and 'm'.
4948 in base 51 = 1k1 and consists of only the digits '1' and 'k'.

The number 4948 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A097545: Numerators of "Farey fraction" approximations to Pi.
A101135: a(1)=1. a(n) = a(n-1) + sum of the squares which are among the first (n-1) terms of the sequence.
A138854: Numbers which are the sum of three cubes of distinct primes.
A163937: Triangle related to the o.g.f.s. of the right hand columns of A028421 (E(x,m=2,n)).
A179186: Numbers n such that phi(n) = phi(n+4), with Euler's totient function phi=A000010.
A190402: Number n for which phi(n) = phi(n'), where phi is the Euler totient function and n' the arithmetic derivative of n.
A227025: T(n,k)=Number of nXk (0,1,2) arrays of permanents of 2X2 subblocks of some (n+1)X(k+1) binary array with rows and columns of the latter in lexicographically nondecreasing order
A241638: Number of partitions p of n such that (number of even numbers in p) = (number of odd numbers in p).
A269011: T(n,k)=Number of nXk binary arrays with some element plus some horizontally, diagonally or antidiagonally adjacent neighbor totalling two exactly once.
A300120: Number of skew partitions whose quotient diagram is connected and whose numerator has weight n.

Number of the day: 94440

Properties of the number 94440:

94440 = 2³ × 3 × 5 × 787 is the 85331^th composite number and is not squarefree.
94440 has 4 distinct prime factors, 32 divisors, 19 antidivisors and 25152 totatives.
94440 has a semiprime digit sum 21 in base 10.
94440 has a Fibonacci digit sum 21 in base 10.
94440 has a triangular digit sum 21 in base 10.
94440 = 23611² - 23609² = 11807² - 11803² = 7873² - 7867² = 4727² - 4717² = 3941² - 3929² = 2371² - 2351² = 1589² - 1559² = 817² - 757² is the difference of 2 nonnegative squares in 8 ways.
94440 is the sum of 2 positive triangular numbers.
94440 is the difference of 2 positive pentagonal numbers in 2 ways.
94440 = 10² + 56² + 302² is the sum of 3 positive squares.
94440² = 56664² + 75552² is the sum of 2 positive squares in 1 way.
94440² is the sum of 3 positive squares.
94440 is a proper divisor of 379⁶ - 1.
94440 is palindromic in (at least) the following bases: 86, 89, and 95.

The number 94440 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A057280: Coefficient triangle of polynomials (rising powers) related to Fibonacci convolutions. Companion triangle to A057995.
A057282: Coefficient triangle of polynomials (falling powers) related to Fibonacci convolutions. Companion triangle to A057281.
A150180: Number of walks within N^3 (the first octant of Z^3) starting at (0,0,0) and consisting of n steps taken from {(-1, -1, 1), (-1, 1, -1), (0, 1, 1), (1, -1, -1), (1, 0, 1)}
A208127: Cardinality of the set f^n({s}), where f is a variant of the Collatz function that replaces any element x in the argument set by both x/2 and 3*x+1, and s is an arbitrary irrational number.
A254942: Number of length n 1..(3+1) arrays with every leading partial sum divisible by 2, 3, 5, 7 or 11

Number of the day: 368560735

George Pólya was born on this day 131 years ago.

Properties of the number 368560735:

368560735 is a cyclic number.
368560735 = 5 × 73712147 is semiprime and squarefree.
368560735 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 294848584 totatives.
368560735 has a prime digit sum 43 in base 10.
Reversing the decimal digits of 368560735 results in a prime.
368560735 = 184280368² - 184280367² = 36856076² - 36856071² is the difference of 2 nonnegative squares in 2 ways.
368560735 is the difference of 2 positive pentagonal numbers in 2 ways.
368560735 is not the sum of 3 positive squares.
368560735² = 221136441² + 294848588² is the sum of 2 positive squares in 1 way.
368560735² is the sum of 3 positive squares.
368560735 is a proper divisor of 31^36856073 - 1.
368560735 = '36856073' + '5' is the concatenation of 2 prime numbers.
368560735 = '3685607' + '35' is the concatenation of 2 semiprime numbers.
368560735 = '3685' + '60735' is the concatenation of 2 sphenic numbers.
368560735 is an emirpimes in (at least) the following bases: 11, 13, 19, 24, 25, 37, 45, 46, 47, 51, 53, 63, 64, 73, 77, 79, 85, 87, 92, and 93.

Number of the day: 958492

Properties of the number 958492:

958492 = 2² × 239623 is the 882982^th composite number and is not squarefree.
958492 has 2 distinct prime factors, 6 divisors, 21 antidivisors and 479244 totatives.
958492 has an emirp digit sum 37 in base 10.
Reversing the decimal digits of 958492 results in a prime.
958492 = 239624² - 239622² is the difference of 2 nonnegative squares in 1 way.
958492 is the difference of 2 positive pentagonal numbers in 1 way.
958492 is not the sum of 3 positive squares.
958492² is the sum of 3 positive squares.
958492 is a proper divisor of 17³⁹⁹³⁷ - 1.

The number 958492 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A257206: Number of (n+2)X(5+2) 0..1 arrays with every 3X3 subblock sum of the two sums of the diagonal and antidiagonal minus the two minimums of the central column and central row nondecreasing horizontally and vertically
A257208: Number of (n+2)X(7+2) 0..1 arrays with every 3X3 subblock sum of the two sums of the diagonal and antidiagonal minus the two minimums of the central column and central row nondecreasing horizontally and vertically

Number of the day: 34260190

Properties of the number 34260190:

34260190 = 2 × 5 × 431 × 7949 is the 32155960^th composite number and is squarefree.
34260190 has 4 distinct prime factors, 16 divisors, 13 antidivisors and 13670560 totatives.
34260190 has a semiprime digit sum 25 in base 10.
34260190 is the difference of 2 positive pentagonal numbers in 3 ways.
34260190 = 445² + 474² + 5817² is the sum of 3 positive squares.
34260190² = 23700690² + 24739400² = 4116912² + 34011934² = 5571106² + 33804192² = 20556114² + 27408152² is the sum of 2 positive squares in 4 ways.
34260190² is the sum of 3 positive squares.
34260190 is a proper divisor of 863⁷⁹⁴⁸ - 1.

Number of the day: 104079

Properties of the number 104079:

104079 = 3 × 34693 is semiprime and squarefree.
104079 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 69384 totatives.
104079 has a semiprime digit sum 21 in base 10.
104079 has a Fibonacci digit sum 21 in base 10.
104079 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 104079 results in an emirpimes.
104079 = 52040² - 52039² = 17348² - 17345² is the difference of 2 nonnegative squares in 2 ways.
104079 is the difference of 2 positive pentagonal numbers in 1 way.
104079 is not the sum of 3 positive squares.
104079² = 61596² + 83895² is the sum of 2 positive squares in 1 way.
104079² is the sum of 3 positive squares.
104079 is a proper divisor of 1489¹² - 1.
104079 = '10407' + '9' is the concatenation of 2 semiprime numbers.
104079 is an emirpimes in (at least) the following bases: 2, 5, 8, 10, 12, 13, 15, 17, 20, 21, 24, 25, 27, 29, 30, 31, 39, 41, 49, 50, 53, 54, 56, 57, 61, 75, 78, and 89.
104079 is palindromic in (at least) the following bases: 93, and -11.

Number of the day: 89568

Properties of the number 89568:

89568 = 2⁵ × 3² × 311 is the 80893^th composite number and is not squarefree.
89568 has 3 distinct prime factors, 36 divisors, 17 antidivisors and 29760 totatives.
89568 has a triangular digit sum 36 in base 10.
89568 is the difference of 2 nonnegative squares in 12 ways.
89568 is the sum of 2 positive triangular numbers.
89568 is the difference of 2 positive pentagonal numbers in 1 way.
89568 = 40² + 52² + 292² is the sum of 3 positive squares.
89568² is the sum of 3 positive squares.
89568 is a proper divisor of 881¹⁰ - 1.

Number of the day: 32383

Properties of the number 32383:

32383 = 13 × 47 × 53 is a sphenic number and squarefree.
32383 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 28704 totatives.
32383 has a prime digit sum 19 in base 10.
Reversing the decimal digits of 32383 results in a semiprime.
32383 = 16192² - 16191² = 1252² - 1239² = 368² - 321² = 332² - 279² is the difference of 2 nonnegative squares in 4 ways.
32383 is the sum of 2 positive triangular numbers.
32383 is the difference of 2 positive pentagonal numbers in 4 ways.
32383 is not the sum of 3 positive squares.
32383² = 17108² + 27495² = 5217² + 31960² = 18800² + 26367² = 12455² + 29892² is the sum of 2 positive squares in 4 ways.
32383² is the sum of 3 positive squares.
32383 is a proper divisor of 659¹² - 1.
32383 = '3' + '2383' is the concatenation of 2 prime numbers.
32383 is palindromic in (at least) the following bases: 8, 32, -2, and -71.
32383 in base 8 = 77177 and consists of only the digits '1' and '7'.
32383 in base 32 = vjv and consists of only the digits 'j' and 'v'.

The number 32383 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A043452: Numbers n such that number of 7's in base 8 is 4.
A049886: a(n)=a(1)+a(2)+...+a(n-1)-a(m), where m=n-1-2^p and 2^p<n-1<=2^(p+1).
A052817: Cycles of rooted trees t where for each t all subtrees at root are distinct. n is total number of nodes.
A070835: Numbers n such that phi(reverse(n)) = sigma(n).
A195460: 2^(2*n+1) - 3*2^n - 1.
A273406: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood.
A273833: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.

Number of the day: 7180

Properties of the number 7180:

7180 = 2² × 5 × 359 is the 6262^th composite number and is not squarefree.
7180 has 3 distinct prime factors, 12 divisors, 7 antidivisors and 2864 totatives.
7180 = 1796² - 1794² = 364² - 354² is the difference of 2 nonnegative squares in 2 ways.
7180 is the difference of 2 positive pentagonal numbers in 2 ways.
7180 = 14² + 30² + 78² is the sum of 3 positive squares.
7180² = 4308² + 5744² is the sum of 2 positive squares in 1 way.
7180² is the sum of 3 positive squares.
7180 is a proper divisor of 719² - 1.
7180 is palindromic in (at least) the following bases: 34, 35, 39, -41, -46, and -74.
7180 in base 23 = dd4 and consists of only the digits '4' and 'd'.
7180 in base 33 = 6jj and consists of only the digits '6' and 'j'.
7180 in base 34 = 676 and consists of only the digits '6' and '7'.
7180 in base 35 = 5u5 and consists of only the digits '5' and 'u'.
7180 in base 38 = 4aa and consists of only the digits '4' and 'a'.
7180 in base 39 = 4S4 and consists of only the digits '4' and 'S'.

The number 7180 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000233: Generalized class numbers.
A022266: a(n) = n*(9*n - 1)/2.
A058625: McKay-Thompson series of class 30d for Monster.
A060093: Number of 5-block ordered bicoverings of an unlabeled n-set.
A105282: Positive integers n such that n^20 + 1 is semiprime (A001358).
A130610: Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.
A134907: a(n) = floor(n*exp(-tan(n))).
A154361: a(n) = 250*n - 70.
A262367: Fixed points of permutations A262323 and A262255.
A285799: Number of partitions of n into parts with an odd number of distinct prime divisors.

Number of the day: 847894

Properties of the number 847894:

847894 = 2 × 19 × 53 × 421 is the 780429^th composite number and is squarefree.
847894 has 4 distinct prime factors, 16 divisors, 23 antidivisors and 393120 totatives.
847894 is the difference of 2 positive pentagonal numbers in 5 ways.
847894 = 2² + 147² + 909² is the sum of 3 positive squares.
847894² = 447944² + 719910² = 397290² + 749056² = 496470² + 687344² = 58406² + 845880² is the sum of 2 positive squares in 4 ways.
847894² is the sum of 3 positive squares.
847894 is a proper divisor of 107³⁰ - 1.

Number of the day: 34801

Nikolai Lobachevsky was born on this day 226 years ago.

Properties of the number 34801:

34801 is a cyclic number.
34801 = 13 × 2677 is semiprime and squarefree.
34801 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 32112 totatives.
Reversing the decimal digits of 34801 results in an emirpimes.
34801 = 17401² - 17400² = 1345² - 1332² is the difference of 2 nonnegative squares in 2 ways.
34801 is the difference of 2 positive pentagonal numbers in 2 ways.
34801 = 49² + 180² = 24² + 185² is the sum of 2 positive squares in 2 ways.
34801 = 4² + 81² + 168² is the sum of 3 positive squares.
34801² = 13385² + 32124² = 8880² + 33649² = 17640² + 29999² = 4745² + 34476² is the sum of 2 positive squares in 4 ways.
34801² is the sum of 3 positive squares.
34801 is a proper divisor of 1033¹² - 1.
34801 = '3' + '4801' is the concatenation of 2 prime numbers.
34801 is an emirpimes in (at least) the following bases: 3, 9, 10, 17, 22, 25, 26, 27, 28, 29, 31, 33, 35, 37, 41, 42, 44, 45, 48, 49, 51, 53, 55, 57, 58, 60, 62, 63, 65, 68, 69, 71, 73, 76, 77, 78, 81, 85, 89, 95, 96, 97, and 100.
34801 is palindromic in (at least) the following bases: 12, -15, and -52.
34801 in base 12 = 18181 and consists of only the digits '1' and '8'.
34801 in base 33 = vvj and consists of only the digits 'j' and 'v'.
34801 in base 51 = DJJ and consists of only the digits 'D' and 'J'.

The number 34801 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A020418: Numbers n such that continued fraction for sqrt(n) has period 79.
A135712: a(n) = (4*n^3 + 11*n^2 + 9*n + 2)/2.
A291543: Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.
A303196: Number of nX7 0..1 arrays with every element equal to 0, 1, 3, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
A303201: Number of 6Xn 0..1 arrays with every element equal to 0, 1, 3, 5 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

Number of the day: 4361400865

Properties of the number 4361400865:

4361400865 is a cyclic number.
4361400865 = 5 × 5419 × 160967 is a sphenic number and squarefree.
4361400865 has 3 distinct prime factors, 8 divisors, 27 antidivisors and 3488455152 totatives.
4361400865 has an emirp digit sum 37 in base 10.
4361400865 = 809³ + 1114³ + 1348³ is the sum of 3 positive cubes in 1 way.
4361400865 = 2180700433² - 2180700432² = 436140089² - 436140084² = 405127² - 399708² = 94031² - 66936² is the difference of 2 nonnegative squares in 4 ways.
4361400865 is the difference of 2 positive pentagonal numbers in 3 ways.
4361400865 = 1025² + 2436² + 65988² is the sum of 3 positive squares.
4361400865² = 2616840519² + 3489120692² is the sum of 2 positive squares in 1 way.
4361400865² is the sum of 3 positive squares.
4361400865 is a proper divisor of 127²³⁵⁵⁶ - 1.

Number of the day: 36779

Properties of the number 36779:

36779 is a cyclic number.
36779 and 36781 form a twin prime pair.
36779 has 13 antidivisors and 36778 totatives.
Reversing the decimal digits of 36779 results in a semiprime.
36779 = 18390² - 18389² is the difference of 2 nonnegative squares in 1 way.
36779 is the difference of 2 positive pentagonal numbers in 1 way.
36779 = 3² + 17² + 191² is the sum of 3 positive squares.
36779² is the sum of 3 positive squares.
36779 is a proper divisor of 1879³⁷ - 1.
36779 = '3' + '6779' is the concatenation of 2 prime numbers.
36779 is an emirp in (at least) the following bases: 7, 9, 11, 13, 14, 19, 29, 31, 37, 43, 46, 49, 51, 57, 59, 69, 73, 79, 84, 91, and 98.
36779 is palindromic in (at least) base 81.
36779 in base 46 = HHP and consists of only the digits 'H' and 'P'.
36779 in base 49 = FFT and consists of only the digits 'F' and 'T'.

The number 36779 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001606: Indices of prime Lucas numbers.
A078848: Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[2,6,4]; short d-string notation of pattern = [264].
A078948: Primes p such that the differences between the 5 consecutive primes starting with p are (2,6,4,2).
A126238: Primes of the form p = prime(n) = (prime(n+3)+prime(n-1))/2.
A130736: Primes n such that n+2, n*(n+2)+12 and n*(n+2)+14 are also prime.
A137476: Prime numbers p such that p^3 - (p+1)^2 and p^3 + (p+1)^2 are both primes.
A216576: Numbers n such that n-th Lucas number is prime, but cannot be written in the form a^2 + 10*b^2.
A226719: a(n) = the first member of a twin prime pair whose sum equals the sums of n consecutive pairs of twin primes.
A250025: Lesser of twin prime pairs of the form (40n - 21, 40n - 19).
A276848: For a lesser p of twin primes, let B_(p+2) and B_p be sequences defined as A159559, but with initial terms p+2 and p respectively. The sequence lists p for which all differences B_(p+2)(n)-B_p(n)<=6.

Number of the day: 29855568

Properties of the number 29855568:

29855568 = 2⁴ × 3 × 107 × 5813 is the 28006194^th composite number and is not squarefree.
29855568 has 4 distinct prime factors, 40 divisors, 19 antidivisors and 9857152 totatives.
29855568 is the difference of 2 nonnegative squares in 12 ways.
29855568 is the sum of 2 positive triangular numbers.
29855568 is the difference of 2 positive pentagonal numbers in 1 way.
29855568 = 4² + 16² + 5464² is the sum of 3 positive squares.
29855568² = 16496832² + 24883920² is the sum of 2 positive squares in 1 way.
29855568² is the sum of 3 positive squares.
29855568 is a proper divisor of 641²⁹⁰⁶ - 1.

Number of the day: 7105633254325

Properties of the number 7105633254325:

7105633254325 = 5² × 17 × 23 × 1789 × 406327 is composite and not squarefree.
7105633254325 has 5 distinct prime factors, 48 divisors, 75 antidivisors and 5114636651520 totatives.
7105633254325 has a semiprime digit sum 46 in base 10.
Reversing the decimal digits of 7105633254325 results in a semiprime.
7105633254325 is the difference of 2 nonnegative squares in 24 ways.
7105633254325 is the sum of 2 positive triangular numbers.
7105633254325 is the difference of 2 positive pentagonal numbers in 23 ways.
7105633254325 = 4620² + 10673² + 2665614² is the sum of 3 positive squares.
7105633254325² = 4722291761300² + 5309424118125² = 592178938165² + 7080914351280² = 1414164237720² + 6963487879915² = 3774805460236² + 6020038843923² = 3046761337773² + 6419288846564² = 3343827413800² + 6269676400875² = 1086743909485² + 7022037568980² = 3009444672420² + 6436867771565² = 4965583709493² + 5082617668976² = 1454564925003² + 6955161020704² = 2704920870635² + 6570648904680² = 1778452646300² + 6879471646125² = 3633566601363² + 6106326039316² = 4436505729120² + 5550445104715² = 218937520467² + 7102259521244² = 2809684161045² + 6526538045560² = 1668175498500² + 6907040933075² = 3535419939821² + 6163670156172² = 4524727447360² + 5478764958645² = 332522982701² + 7097848435332² = 4263379952595² + 5684506603460² = 1989577311211² + 6821407924152² is the sum of 2 positive squares in 22 ways.
7105633254325² is the sum of 3 positive squares.
7105633254325 is a proper divisor of 919^4063260 - 1.

Number of the day: 53430

Properties of the number 53430:

53430 = 2 × 3 × 5 × 13 × 137 is the 47981^th composite number and is squarefree.
53430 has 5 distinct prime factors, 32 divisors, 15 antidivisors and 13056 totatives.
53430 has an emirpimes digit sum 15 in base 10.
53430 has a triangular digit sum 15 in base 10.
53430 = 58² + … + 70² is the sum of at least 2 consecutive positive squares in 1 way.
53430 is the sum of 2 positive triangular numbers.
53430 is the difference of 2 positive pentagonal numbers in 2 ways.
53430 = 26² + 35² + 227² is the sum of 3 positive squares.
53430² = 34320² + 40950² = 15930² + 51000² = 24600² + 47430² = 17856² + 50358² = 2886² + 53352² = 23184² + 48138² = 12168² + 52026² = 8778² + 52704² = 31242² + 43344² = 32058² + 42744² = 13152² + 51786² = 27126² + 46032² = 20550² + 49320² is the sum of 2 positive squares in 13 ways.
53430² is the sum of 3 positive squares.
53430 is a proper divisor of 311⁴ - 1.
53430 = '534' + '30' is the concatenation of 2 sphenic numbers.
53430 in base 37 = 1212 and consists of only the digits '1' and '2'.

The number 53430 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A024205: [ (4th elementary symmetric function of S(n))/(2nd elementary symmetric function of S(n)) ], where S(n) = {first n+3 odd positive integers}.
A072289: One eighty-fourth the area of primitive Pythagorean triangles with (increasing) square hypotenuses (precisely those of A008846).
A077521: a(1) = 1, a(n) = smallest squarefree number which begins with n and is a multiple of the previous term.
A111696: a(n) = C(n,a)+C(n,b)+C(n,c)... where n = abc... are the decimal digits of n.
A123975: Number of Garden of Eden partitions of n in Bulgarian Solitaire.
A274062: Even numbers such that the sum of the odd divisors is a Fibonacci number F and the sum of the even divisors is 2F.

Number of the day: 876266

Properties of the number 876266:

876266 = 2 × 438133 is semiprime and squarefree.
876266 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 438132 totatives.
876266 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 876266 results in an emirpimes.
876266 = 115² + 929² is the sum of 2 positive squares in 1 way.
876266 = 59² + 116² + 927² is the sum of 3 positive squares.
876266² = 213670² + 849816² is the sum of 2 positive squares in 1 way.
876266² is the sum of 3 positive squares.
876266 is a proper divisor of 43¹²⁵⁹ - 1.
876266 = '8762' + '66' is the concatenation of 2 sphenic numbers.
876266 is an emirpimes in (at least) the following bases: 2, 4, 6, 7, 9, 10, 16, 20, 21, 24, 28, 30, 34, 39, 40, 41, 43, 48, 50, 53, 62, 64, 65, 66, 67, 72, 76, 77, 80, 82, 83, 85, 86, 88, 89, 91, 93, 97, 98, and 99.

Number of the day: 11232018

John Wallis was born on this day 402 years ago.

Happy Fibonacci Day!

Properties of the number 11232018:

11232018 = 2 × 3² × 7 × 97 × 919 is the 10491197^th composite number and is not squarefree.
11232018 has 5 distinct prime factors, 48 divisors, 31 antidivisors and 3172608 totatives.
11232018 is the sum of 2 positive triangular numbers.
11232018 = 29² + 124² + 3349² is the sum of 3 positive squares.
11232018² = 7526610² + 8337168² is the sum of 2 positive squares in 1 way.
11232018² is the sum of 3 positive squares.
11232018 is a proper divisor of 971⁶ - 1.

Number of the day: 419339

Properties of the number 419339:

419339 = 17² × 1451 is the 384000^th composite number and is not squarefree.
419339 has 2 distinct prime factors, 6 divisors, 17 antidivisors and 394400 totatives.
419339 has a prime digit sum 29 in base 10.
Reversing the decimal digits of 419339 results in a semiprime.
419339 = 209670² - 209669² = 12342² - 12325² = 870² - 581² is the difference of 2 nonnegative squares in 3 ways.
419339 is the difference of 2 positive pentagonal numbers in 2 ways.
419339 = 17² + 55² + 645² is the sum of 3 positive squares.
419339² = 233611² + 348240² = 197336² + 370005² is the sum of 2 positive squares in 2 ways.
419339² is the sum of 3 positive squares.
419339 is a proper divisor of 1021⁸⁵ - 1.
419339 = '4' + '19339' is the concatenation of 2 semiprime numbers.

Number of the day: 7570

Properties of the number 7570:

7570 = 2 × 5 × 757 is a sphenic number and squarefree.
7570 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 3024 totatives.
7570 has a prime digit sum 19 in base 10.
7570 is the difference of 2 positive pentagonal numbers in 2 ways.
7570 = 53² + 69² = 1² + 87² is the sum of 2 positive squares in 2 ways.
7570 = 15² + 28² + 81² is the sum of 3 positive squares.
7570² = 4542² + 6056² = 1952² + 7314² = 174² + 7568² = 4680² + 5950² is the sum of 2 positive squares in 4 ways.
7570² is the sum of 3 positive squares.
7570 is a proper divisor of 1427⁴ - 1.
7570 is palindromic in (at least) the following bases: 3, 27, 39, 47, 87, -28, -86, and -87.
7570 in base 3 = 101101101 and consists of only the digits '0' and '1'.
7570 in base 20 = iia and consists of only the digits 'a' and 'i'.
7570 in base 27 = aaa and consists of only the digit 'a'.
7570 in base 35 = 66a and consists of only the digits '6' and 'a'.
7570 in base 39 = 4c4 and consists of only the digits '4' and 'c'.
7570 in base 43 = 442 and consists of only the digits '2' and '4'.
7570 in base 46 = 3QQ and consists of only the digits '3' and 'Q'.
7570 in base 47 = 3K3 and consists of only the digits '3' and 'K'.
7570 in base 61 = 226 and consists of only the digits '2' and '6'.

The number 7570 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A007107: Number of labeled 2-regular digraphs with n nodes.
A048885: Number of nonisomorphic orthogonal arrays OA(4n,4n-1,2,2).
A060879: Intrinsic 9-palindromes: n is an intrinsic k-palindrome if it is a k-digit palindrome in some base.
A069894: Centered square numbers: a(n) = 4*n^2 + 4*n + 2.
A154360: a(n) = 250*n - 180.
A210766: Number of 8-hoops with n symbols and no a-rooted trees.
A252167: T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and column sum equal to 2 3 4 6 or 7 and every 3X3 diagonal and antidiagonal sum not equal to 2 3 4 6 or 7
A264925: G.f.: 1 / Product_{n>=0} (1 - x^(n+5))^((n+1)*(n+2)*(n+3)*(n+4)/4!).
A284990: Triangle T(n,t) read by rows: the number of n X n {0,1} matrices with trace t where each row sum and each column sum is 3.
A299285: Coordination sequence for "tea" 3D uniform tiling.

Number of the day: 3207

Properties of the number 3207:

3207 = 3 × 1069 is semiprime and squarefree.
3207 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 2136 totatives.
3207 has an oblong digit sum 12 in base 10.
Reversing the decimal digits of 3207 results in an emirpimes.
3207 = 1604² - 1603² = 536² - 533² is the difference of 2 nonnegative squares in 2 ways.
3207 is the sum of 2 positive triangular numbers.
3207 is the difference of 2 positive pentagonal numbers in 1 way.
3207 is not the sum of 3 positive squares.
3207² = 2193² + 2340² is the sum of 2 positive squares in 1 way.
3207² is the sum of 3 positive squares.
3207 is a proper divisor of 1889⁴ - 1.
3207 is an emirpimes in (at least) the following bases: 2, 5, 7, 9, 10, 11, 15, 36, 39, 42, 44, 46, 47, 48, 50, 58, 59, 60, 62, 63, 69, 73, 74, 75, 77, 89, 90, 96, and 99.
3207 is palindromic in (at least) the following bases: 17, and -36.
3207 in base 17 = b1b and consists of only the digits '1' and 'b'.
3207 in base 56 = 11F and consists of only the digits '1' and 'F'.

The number 3207 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A014670: G.f.: (1+x)*(1+x^3)*(1+x^5)*(1+x^7)*(1+x^9)/((1-x^2)*(1-x^4)*(1-x^6)*(1-x^8)*(1-x^10)).
A072475: Sum of next n composite numbers.
A124200: Numbers n such that 1 + n + n^3 + n^5 + n^7 + n^9 + n^11 + n^13 + n^15 + n^17 + n^19 + n^21 + n^23 + n^25 + n^27 + n^29 + n^31 + n^33 + n^35 + n^37 + n^39 + n^41 + n^43 is prime.
A137994: a(n) is the smallest integer > a(n-1) such that {Pi^a(n)} < {Pi^a(n-1)}, where {x} = x - floor(x), a(1)=1.
A207542: Number of solid standard Young tableaux with n cells.
A243577: Integers of the form 8k+7 that can be written as a sum of four distinct 'almost consecutive' squares.
A243580: Integers of the form 8k + 7 that can be written as a sum of four distinct squares of the form m, m + 1, m + 3, m + 5, where m == 2 (mod 4).
A293403: Greatest integer k such that k/n^2 < (3 + sqrt(5))/2.
A293405: The integer k that minimizes |k/n^2 - tau^2|, where tau = (1+sqrt(5))/2 (golden ratio).
A298791: Partial sums of A298789.

Number of the day: 5023

August Ferdinand Möbius was born on this day 228 years ago.

Properties of the number 5023:

5023 is a cyclic number.
5021 and 5023 form a twin prime pair.
5023 has 17 antidivisors and 5022 totatives.
5023 has a semiprime digit sum 10 in base 10.
5023 has a triangular digit sum 10 in base 10.
Reversing the decimal digits of 5023 results in a semiprime.
5023 = 2512² - 2511² is the difference of 2 nonnegative squares in 1 way.
5023 is the difference of 2 positive pentagonal numbers in 1 way.
5023 is not the sum of 3 positive squares.
5023² is the sum of 3 positive squares.
5023 is a proper divisor of 953³ - 1.
5023 is an emirp in (at least) the following bases: 2, 7, 13, 14, 16, 17, 26, 33, 35, 40, 41, 49, 52, 55, 56, 59, 61, 64, 65, 66, 68, 70, 71, 74, 77, 79, 82, 85, 88, 89, 91, 94, and 95.
5023 is palindromic in (at least) the following bases: 23, 54, 62, -29, -81, and -93.
5023 in base 23 = 9b9 and consists of only the digits '9' and 'b'.
5023 in base 28 = 6bb and consists of only the digits '6' and 'b'.
5023 in base 53 = 1ff and consists of only the digits '1' and 'f'.
5023 in base 54 = 1d1 and consists of only the digits '1' and 'd'.
5023 in base 61 = 1LL and consists of only the digits '1' and 'L'.
5023 in base 62 = 1J1 and consists of only the digits '1' and 'J'.

The number 5023 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A031934: Lower prime of a pair of consecutive primes having a difference of 16.
A045711: Primes with first digit 5.
A046257: a(1) = 7; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A074982: Primes for which the three closest primes are smaller.
A119617: Integers of the form c(n)/b(n) where c(n+1)=c(n)+(n+1)^4 with c(0)=1 and b(n+1)=b(n)+(n+1)^2 with b(0)=1.
A127578: Primes congruent to 31 mod 32.
A134153: a(n) = 15n^2 + 9n + 1.
A142005: Primes congruent to 1 mod 31.
A153377: Larger of two consecutive prime numbers such that p1*p2*d+d=average of twin prime pairs, d (delta)=p2-p1.
A297410: Where prime race among 11n+1, ..., 11n+10 changes leader.

Number of the day: 1356

Properties of the number 1356:

1356 = 2² × 3 × 113 is the 1138^th composite number and is not squarefree.
1356 has 3 distinct prime factors, 12 divisors, 3 antidivisors and 448 totatives.
1356 has an emirpimes digit sum 15 in base 10.
1356 has a triangular digit sum 15 in base 10.
1356 has an oblong digit product 90 in base 10.
1356 has sum of divisors equal to 3192 which is an oblong number.
Reversing the decimal digits of 1356 results in a sphenic number.
1356 = 8² + … + 16² is the sum of at least 2 consecutive positive squares in 1 way.
1356 = 340² - 338² = 116² - 110² is the difference of 2 nonnegative squares in 2 ways.
1356 is the difference of 2 positive pentagonal numbers in 1 way.
1356 = 14² + 22² + 26² is the sum of 3 positive squares.
1356² = 180² + 1344² is the sum of 2 positive squares in 1 way.
1356² is the sum of 3 positive squares.
1356 is a proper divisor of 227² - 1.
1356 is palindromic in (at least) the following bases: 15, and -15.
1356 in base 14 = 6cc and consists of only the digits '6' and 'c'.
1356 in base 15 = 606 and consists of only the digits '0' and '6'.
1356 in base 36 = 11o and consists of only the digits '1' and 'o'.

The number 1356 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001934: Expansion of 1/theta_4(q)^2 in powers of q.
A007534: Even numbers that are not the sum of a pair of twin primes.
A018039: Powers of cube root of 22 rounded down.
A018040: Powers of cube root of 22 rounded to nearest integer.
A046741: Triangle read by rows giving number of arrangements of k dumbbells on 2 X n grid (n >= 0, k >= 0).
A111871: Prime gaps q-p with n-th record merit referred to in A111870.
A162532: Numbers n such that their largest divisor <= sqrt(n) equals 12.
A215861: Number T(n,k) of simple labeled graphs on n nodes with exactly k connected components that are trees or cycles; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A250474: Number of times prime(n) occurs as the least prime factor among numbers 1 .. prime(n)^3: a(n) = A078898(A030078(n)).
A298118: Number of unlabeled rooted trees with n nodes in which all positive outdegrees are odd.

Number of the day: 99678

Properties of the number 99678:

99678 = 2 × 3 × 37 × 449 is the 90115^th composite number and is squarefree.
99678 has 4 distinct prime factors, 16 divisors, 13 antidivisors and 32256 totatives.
99678 has an emirpimes digit sum 39 in base 10.
99678 is the difference of 2 positive pentagonal numbers in 3 ways.
99678 = 22² + 35² + 313² is the sum of 3 positive squares.
99678² = 62160² + 77922² = 53550² + 84072² = 33528² + 93870² = 32328² + 94290² is the sum of 2 positive squares in 4 ways.
99678² is the sum of 3 positive squares.
99678 is a proper divisor of 541²⁴ - 1.
99678 is palindromic in (at least) base -11.

The number 99678 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A064262: Values of m such that N=(am+1)(bm+1)(cm+1) is a Carmichael number, where a,b,c = 1,2,51.
A248534: Number of length n+3 0..5 arrays with every four consecutive terms having the sum of some three elements equal to three times the fourth.

Number of the day: 544978

Properties of the number 544978:

544978 = 2 × 7² × 67 × 83 is the 500029^th composite number and is not squarefree.
544978 has 4 distinct prime factors, 24 divisors, 23 antidivisors and 227304 totatives.
544978 has an emirp digit sum 37 in base 10.
Reversing the decimal digits of 544978 results in a sphenic number.
544978 is the difference of 2 positive pentagonal numbers in 9 ways.
544978 = 9² + 116² + 729² is the sum of 3 positive squares.
544978² is the sum of 3 positive squares.
544978 is a proper divisor of 499⁴² - 1.

Number of the day: 785039

Properties of the number 785039:

785039 is a cyclic number.
785039 = 67 × 11717 is semiprime and squarefree.
785039 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 773256 totatives.
Reversing the decimal digits of 785039 results in a sphenic number.
785039 = 392520² - 392519² = 5892² - 5825² is the difference of 2 nonnegative squares in 2 ways.
785039 is the difference of 2 positive pentagonal numbers in 2 ways.
785039 is not the sum of 3 positive squares.
785039² = 51255² + 783364² is the sum of 2 positive squares in 1 way.
785039² is the sum of 3 positive squares.
785039 is a proper divisor of 401²⁰² - 1.
785039 = '78503' + '9' is the concatenation of 2 semiprime numbers.
785039 is an emirpimes in (at least) the following bases: 5, 6, 11, 13, 14, 15, 16, 17, 22, 23, 31, 32, 34, 39, 43, 44, 46, 47, 49, 55, 59, 61, 63, 64, 67, 69, 70, 72, 74, 75, 76, 77, 81, 92, 94, 95, 96, 98, and 100.

Number of the day: 354515

Properties of the number 354515:

354515 = 5 × 7² × 1447 is the 324176^th composite number and is not squarefree.
354515 has 3 distinct prime factors, 12 divisors, 17 antidivisors and 242928 totatives.
354515 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 354515 results in a sphenic number.
354515 = 177258² - 177257² = 35454² - 35449² = 25326² - 25319² = 5082² - 5047² = 3642² - 3593² = 846² - 601² is the difference of 2 nonnegative squares in 6 ways.
354515 is the difference of 2 positive pentagonal numbers in 6 ways.
354515 = 29² + 45² + 593² is the sum of 3 positive squares.
354515² = 212709² + 283612² is the sum of 2 positive squares in 1 way.
354515² is the sum of 3 positive squares.
354515 is a proper divisor of 743⁸⁴ - 1.
354515 = '3545' + '15' is the concatenation of 2 semiprime numbers.
354515 is palindromic in (at least) base -94.

The number 354515 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A224670: Number of (n+1) X 3 0..2 matrices with each 2 X 2 subblock idempotent
A278544: Arithmetic derivative of the prime-factorization representation of the n-th Stern polynomial: a(n) = A003415(A260443(n)).