Number of the day: 167

Richard K. Guy was born on this day 104 years ago.

Properties of the number 167:

167 is a cyclic number.
167 is the 39^th prime.
167 has 7 antidivisors and 166 totatives.
167 has a semiprime digit sum 14 in base 10.
167 has a sphenic digit product 42 in base 10.
167 has an oblong digit product 42 in base 10.
Reversing the decimal digits of 167 results in an emirp.
167 = 84² - 83² is the difference of 2 nonnegative squares in 1 way.
167 is the difference of 2 positive pentagonal numbers in 1 way.
167 is not the sum of 3 positive squares.
167² is the sum of 3 positive squares.
167 is a proper divisor of 1669² - 1.
167 is an emirp in (at least) the following bases: 2, 3, 5, 7, 10, 12, 14, 15, 19, 22, 25, 28, 29, 32, 33, 39, 41, 43, 44, 49, 50, 55, 57, 67, 69, 71, 75, 85, 86, 87, 91, 93, 94, 95, 97, 98, 99, and 100.
167 is palindromic in (at least) the following bases: -6, -11, and -83.
167 in base 12 = 11b and consists of only the digits '1' and 'b'.

The number 167 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000040: The prime numbers.
A000837: Number of partitions of n into relatively prime parts. Also aperiodic partitions.
A000961: Powers of primes. Alternatively, 1 and the prime powers (p^k, p prime, k >= 1).
A002145: Primes of the form 4*k+3.
A004767: a(n) = 4*n + 3.
A005385: Safe primes p: (p-1)/2 is also prime.
A006567: Emirps (primes whose reversal is a different prime).
A008578: Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime).
A050376: "Fermi-Dirac primes": numbers of the form p^(2^k) where p is prime and k >= 0.
A065091: Odd primes.

Number of the day: 164193

Properties of the number 164193:

164193 = 3 × 229 × 239 is a sphenic number and squarefree.
164193 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 108528 totatives.
164193 = 82097² - 82096² = 27367² - 27364² = 473² - 244² = 463² - 224² is the difference of 2 nonnegative squares in 4 ways.
164193 is the difference of 2 positive pentagonal numbers in 1 way.
164193 = 4² + 31² + 404² is the sum of 3 positive squares.
164193² = 43020² + 158457² is the sum of 2 positive squares in 1 way.
164193² is the sum of 3 positive squares.
164193 is a proper divisor of 1697⁴² - 1.
164193 = '1641' + '93' is the concatenation of 2 emirpimes.

Number of the day: 4566

Properties of the number 4566:

4566 is the 1156^th totient number.
4566 = 2 × 3 × 761 is a sphenic number and squarefree.
4566 has 3 distinct prime factors, 8 divisors, 5 antidivisors and 1520 totatives.
4566 has a semiprime digit sum 21 in base 10.
4566 has a Fibonacci digit sum 21 in base 10.
4566 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 4566 results in a sphenic number.
4566 is the sum of 2 positive triangular numbers.
4566 is the difference of 2 positive pentagonal numbers in 2 ways.
4566 = 26² + 41² + 47² is the sum of 3 positive squares.
4566² = 234² + 4560² is the sum of 2 positive squares in 1 way.
4566² is the sum of 3 positive squares.
4566 is a proper divisor of 1523² - 1.
4566 = '4' + '566' is the concatenation of 2 semiprime numbers.
4566 = '45' + '66' is the concatenation of 2 triangular numbers.
4566 is palindromic in (at least) the following bases: 39, 55, -30, -39, and -83.
4566 in base 19 = cc6 and consists of only the digits '6' and 'c'.
4566 in base 22 = 99c and consists of only the digits '9' and 'c'.
4566 in base 25 = 77g and consists of only the digits '7' and 'g'.
4566 in base 38 = 366 and consists of only the digits '3' and '6'.
4566 in base 39 = 303 and consists of only the digits '0' and '3'.
4566 in base 54 = 1UU and consists of only the digits '1' and 'U'.
4566 in base 55 = 1S1 and consists of only the digits '1' and 'S'.

The number 4566 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A018783: Number of partitions of n into parts having a common factor.
A047968: a(n) = Sum_{d|n} p(d), where p(d) = A000041 = number of partitions of d.
A161704: a(n) = (3*n^5 - 35*n^4 + 145*n^3 - 235*n^2 + 152*n + 30)/30.
A182819: G.f.: exp( Sum_{n>=1} sigma(3n)*x^n/n ).
A222986: T(n,k)=Number of nXk 0..2 arrays with no more than floor(nXk/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..2 order
A279787: Twice-partitioned numbers where the first partition is constant.
A299257: Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 3.12.12 2D tiling (cf. A250122).
A305551: Number of partitions of partitions of n where all partitions have the same sum.
A317715: Number of ways to split an integer partition of n into consecutive subsequences with equal sums.
A330570: Partial sums of A097988 (d_3(n)^2).

Number of the day: 37348919

Properties of the number 37348919:

37348919 is a cyclic number.
37348919 = 593 × 62983 is semiprime and squarefree.
37348919 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 37285344 totatives.
Reversing the decimal digits of 37348919 results in a sphenic number.
37348919 = 18674460² - 18674459² = 31788² - 31195² is the difference of 2 nonnegative squares in 2 ways.
37348919 is the difference of 2 positive pentagonal numbers in 2 ways.
37348919 is not the sum of 3 positive squares.
37348919² = 23177744² + 29287095² is the sum of 2 positive squares in 1 way.
37348919² is the sum of 3 positive squares.
37348919 is a proper divisor of 1187³¹⁴⁹¹ - 1.
37348919 = '37' + '348919' is the concatenation of 2 prime numbers.
37348919 = '3734891' + '9' is the concatenation of 2 semiprime numbers.
37348919 is an emirpimes in (at least) the following bases: 2, 4, 5, 16, 22, 23, 25, 32, 36, 37, 38, 39, 43, 47, 51, 52, 55, 56, 58, 59, 64, 66, 70, 71, 72, 73, 76, 77, 78, 82, 90, 98, and 99.

Number of the day: 61701

Properties of the number 61701:

61701 = 3 × 131 × 157 is a sphenic number and squarefree.
61701 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 40560 totatives.
61701 has an emirpimes digit sum 15 in base 10.
61701 has a triangular digit sum 15 in base 10.
61701 = 30851² - 30850² = 10285² - 10282² = 301² - 170² = 275² - 118² is the difference of 2 nonnegative squares in 4 ways.
61701 is the sum of 2 positive triangular numbers.
61701 is the difference of 2 positive pentagonal numbers in 1 way.
61701 = 1² + 14² + 248² is the sum of 3 positive squares.
61701² = 33405² + 51876² is the sum of 2 positive squares in 1 way.
61701² is the sum of 3 positive squares.
61701 is a proper divisor of 1571² - 1.
61701 = '61' + '701' is the concatenation of 2 prime numbers.
61701 is palindromic in (at least) base -41.

The number 61701 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A292760: E.g.f.: (tan x + sec x)*(E.g.f. for A000738).

Number of the day: 259278

Properties of the number 259278:

259278 = 2 × 3 × 79 × 547 is the 236495^th composite number and is squarefree.
259278 has 4 distinct prime factors, 16 divisors, 19 antidivisors and 85176 totatives.
259278 has a semiprime digit sum 33 in base 10.
259278 = 1² + 14² + 509² is the sum of 3 positive squares.
259278² is the sum of 3 positive squares.
259278 is a proper divisor of 719²⁶ - 1.
259278 = '25' + '9278' is the concatenation of 2 semiprime numbers.

Number of the day: 79178117

Gerolamo Cardano was born on this day 519 years ago.

Properties of the number 79178117:

79178117 is a cyclic number.
79178117 and 79178119 form a twin prime pair.
79178117 has 13 antidivisors and 79178116 totatives.
79178117 has a prime digit sum 41 in base 10.
Reversing the decimal digits of 79178117 results in a semiprime.
79178117 = 39589059² - 39589058² is the difference of 2 nonnegative squares in 1 way.
79178117 is the difference of 2 positive pentagonal numbers in 1 way.
79178117 = 5329² + 7126² is the sum of 2 positive squares in 1 way.
79178117 = 95² + 174² + 8896² is the sum of 3 positive squares.
79178117² = 22381635² + 75948908² is the sum of 2 positive squares in 1 way.
79178117² is the sum of 3 positive squares.
79178117 is a proper divisor of 37^19794529 - 1.
79178117 = '79' + '178117' is the concatenation of 2 prime numbers.
79178117 is an emirp in (at least) the following bases: 5, 13, 19, 20, 21, 27, 31, 34, 37, 43, 59, 61, 73, 79, 83, 87, 88, 92, and 100.
79178117 in base 3 = 12111222122221212 and consists of only the digits '1' and '2'.

Number of the day: 7583078565

Properties of the number 7583078565:

7583078565 = 3² × 5 × 17 × 9912521 is the 7233600160^th composite number and is not squarefree.
7583078565 has 4 distinct prime factors, 24 divisors, 23 antidivisors and 3806407680 totatives.
7583078565 = 244³ + 1440³ + 1661³ is the sum of 3 positive cubes in 1 way.
7583078565 is the difference of 2 nonnegative squares in 12 ways.
7583078565 is the difference of 2 positive pentagonal numbers in 4 ways.
7583078565 = 46671² + 73518² = 34878² + 79791² = 42906² + 75777² = 6774² + 86817² is the sum of 2 positive squares in 4 ways.
7583078565 = 542² + 745² + 87076² is the sum of 3 positive squares.
7583078565² = 4549847139² + 6066462852² = 2487499092² + 7163478819² = 382225059² + 7573439412² = 3901228893² + 6502575924² = 3211656804² + 6869377053² = 1176196716² + 7491304413² = 5150128797² + 5565900996² = 1159764957² + 7493865876² = 3226714083² + 6862317156² = 3568507560² + 6690951675² = 3553825275² + 6698761560² = 780562440² + 7542798075² = 4238283600² + 6288086565² is the sum of 2 positive squares in 13 ways.
7583078565² is the sum of 3 positive squares.
7583078565 is a proper divisor of 593^1982504 - 1.
7583078565 = '75830785' + '65' is the concatenation of 2 semiprime numbers.
7583078565 = '75830' + '78565' is the concatenation of 2 sphenic numbers.

Number of the day: 550

Properties of the number 550:

550 = 2 × 5² × 11 is the 448^th composite number and is not squarefree.
550 has 3 distinct prime factors, 12 divisors, 9 antidivisors and 200 totatives.
550 has a semiprime digit sum 10 in base 10.
550 has a triangular digit sum 10 in base 10.
550 = (4 × 5)/2 + … + (14 × 15)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
550 is the difference of 2 positive pentagonal numbers in 2 ways.
550 = 3² + 10² + 21² is the sum of 3 positive squares.
550² = 330² + 440² = 154² + 528² is the sum of 2 positive squares in 2 ways.
550² is the sum of 3 positive squares.
550 is a proper divisor of 199² - 1.
550 = '55' + '0' is the concatenation of 2 triangular numbers.
550 = '55' + '0' is the concatenation of 2 Fibonacci numbers.
550 is palindromic in (at least) the following bases: 24, 49, 54, -13, and -61.
550 in base 7 = 1414 and consists of only the digits '1' and '4'.
550 consists of only the digits '0' and '5'.
550 in base 13 = 334 and consists of only the digits '3' and '4'.
550 in base 16 = 226 and consists of only the digits '2' and '6'.

The number 550 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002411: Pentagonal pyramidal numbers: a(n) = n^2*(n+1)/2.
A003349: Numbers that are the sum of 4 positive 5th powers.
A008593: Multiples of 11.
A028552: a(n) = n*(n+3).
A052224: Numbers whose sum of digits is 10.
A191113: Increasing sequence generated by these rules:a(1)=1, and if x is in a then 3x-2 and 4x-2 are in a.
A194368: Numbers n such that Sum_{k=1..n} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(2) and < > denotes fractional part.
A299281: Coordination sequence for "reo-e" 3D uniform tiling.
A306017: Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.
A333218: Numbers k such that the k-th composition in standard order is a permutation (of an initial interval).

Number of the day: 12768

Properties of the number 12768:

12768 is the 2976^th totient number.
12768 = 2⁵ × 3 × 7 × 19 is the 11244^th composite number and is not squarefree.
12768 has 4 distinct prime factors, 48 divisors, 9 antidivisors and 3456 totatives.
Reversing the decimal digits of 12768 results in a sphenic number.
12768 is the difference of 2 nonnegative squares in 16 ways.
12768 = 8² + 52² + 100² is the sum of 3 positive squares.
12768² is the sum of 3 positive squares.
12768 is a proper divisor of 113² - 1.
12768 is palindromic in (at least) the following bases: 15, -40, and -69.
12768 in base 15 = 3bb3 and consists of only the digits '3' and 'b'.
12768 in base 23 = 1133 and consists of only the digits '1' and '3'.
12768 in base 39 = 8FF and consists of only the digits '8' and 'F'.
12768 in base 50 = 55I and consists of only the digits '5' and 'I'.
12768 in base 56 = 440 and consists of only the digits '0' and '4'.

The number 12768 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A020342: Vampire numbers: (definition 1): n has a nontrivial factorization using n's digits.
A046763: Numbers n such that the sum of the cubes of the divisors of n is divisible by n.
A059283: Triangle T(n,k) (0<= k <=n) read by rows. Left edge is 1, 0, 0, ... Otherwise each entry is sum of entry to left, entries immediately above it to left and right and entry directly above it 2 rows back.
A071943: Triangle T(n,k) (n>=0, 0 <= k <= n) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R=(1,0), V=(0,1) and D=(1,2).
A084920: a(n) = (prime(n)-1)*(prime(n)+1).
A101135: a(1)=1. a(n) = a(n-1) + sum of the squares which are among the first (n-1) terms of the sequence.
A101427: Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.
A139041: Sum of divisors of the number of partitions of n.
A152751: 3 times octagonal numbers: 3*n*(3*n-2).
A187238: Numbers divisible by at least four of their digits, different and >1.

Number of the day: 645425

Properties of the number 645425:

645425 = 5² × 11 × 2347 is the 592950^th composite number and is not squarefree.
645425 has 3 distinct prime factors, 12 divisors, 19 antidivisors and 469200 totatives.
645425 has an emirpimes digit sum 26 in base 10.
645425 = 322713² - 322712² = 64545² - 64540² = 29343² - 29332² = 12921² - 12896² = 5895² - 5840² = 1311² - 1036² is the difference of 2 nonnegative squares in 6 ways.
645425 is the difference of 2 positive pentagonal numbers in 6 ways.
645425 = 14² + 45² + 802² is the sum of 3 positive squares.
645425² = 387255² + 516340² = 180719² + 619608² is the sum of 2 positive squares in 2 ways.
645425² is the sum of 3 positive squares.
645425 is a proper divisor of 1063⁶⁰ - 1.

Number of the day: 87790013

Ky Fan was born on this day 106 years ago.

Properties of the number 87790013:

87790013 is a cyclic number.
87790013 = 19 × 83 × 179 × 311 is the 82693698^th composite number and is squarefree.
87790013 has 4 distinct prime factors, 16 divisors, 39 antidivisors and 81445680 totatives.
87790013 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 87790013 results in a semiprime.
87790013 = 43895007² - 43895006² = 2310273² - 2310254² = 528897² - 528814² = 245313² - 245134² = 141297² - 140986² = 28623² - 27046² = 14607² - 11206² = 10383² - 4474² is the difference of 2 nonnegative squares in 8 ways.
87790013 is the difference of 2 positive pentagonal numbers in 8 ways.
87790013 = 109² + 306² + 9364² is the sum of 3 positive squares.
87790013² is the sum of 3 positive squares.
87790013 is a proper divisor of 331¹⁶⁵⁵⁴ - 1.
87790013 = '87' + '790013' is the concatenation of 2 semiprime numbers.

Number of the day: 4220

Adrien-Marie Legendre was born on this day 268 years ago.

Properties of the number 4220:

4220 is the 1074^th totient number.
4220 = 2² × 5 × 211 is the 3641^th composite number and is not squarefree.
4220 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 1680 totatives.
4220 has a Fibonacci digit sum 8 in base 10.
4220 = 1056² - 1054² = 216² - 206² is the difference of 2 nonnegative squares in 2 ways.
4220 is the difference of 2 positive pentagonal numbers in 2 ways.
4220 is not the sum of 3 positive squares.
4220² = 2532² + 3376² is the sum of 2 positive squares in 1 way.
4220² is the sum of 3 positive squares.
4220 is a proper divisor of 421² - 1.
4220 = '42' + '20' is the concatenation of 2 oblong numbers.
4220 is palindromic in (at least) the following bases: 31, 38, -34, and -57.
4220 in base 24 = 77k and consists of only the digits '7' and 'k'.
4220 in base 26 = 668 and consists of only the digits '6' and '8'.
4220 in base 30 = 4kk and consists of only the digits '4' and 'k'.
4220 in base 31 = 4c4 and consists of only the digits '4' and 'c'.
4220 in base 37 = 332 and consists of only the digits '2' and '3'.
4220 in base 38 = 2Z2 and consists of only the digits '2' and 'Z'.

The number 4220 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A028931: Strings giving winning positions in Tchoukaillon (or Mancala) solitaire.
A077588: Maximum number of regions into which the plane is divided by n triangles.
A108643: Number of binary rooted trees with n nodes and internal path length n.
A124353: Number of (directed) Hamiltonian circuits on the n-antiprism graph.
A134212: Positions of 12 after decimal point in decimal expansion of Pi.
A205590: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock having the same number of equal edges as its horizontal or vertical neighbors, and new values 0..2 introduced in row major order.
A276550: Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.
A283726: T(n,k)=Number of nXk 0..1 arrays with no 1 equal to more than two of its horizontal, diagonal and antidiagonal neighbors, with the exception of exactly one element.
A325583: G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 3*x)^n - A(x))^(n+1), where A(0) = 0.
A333139: The number of regions inside a decagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Number of the day: 2410

Bernhard Riemann was born on this day 194 years ago.

Properties of the number 2410:

2410 is the 646^th totient number.
2410 = 2 × 5 × 241 is a sphenic number and squarefree.
2410 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 960 totatives.
2410 has a prime digit sum 7 in base 10.
2410 is the difference of 2 positive pentagonal numbers in 2 ways.
2410 = 27² + 41² = 3² + 49² is the sum of 2 positive squares in 2 ways.
2410 = 5² + 33² + 36² is the sum of 3 positive squares.
2410² = 1446² + 1928² = 952² + 2214² = 294² + 2392² = 1200² + 2090² is the sum of 2 positive squares in 4 ways.
2410² is the sum of 3 positive squares.
2410 is a proper divisor of 659⁴ - 1.
2410 is palindromic in (at least) the following bases: 15, -16, -29, -43, and -73.
2410 in base 4 = 211222 and consists of only the digits '1' and '2'.
2410 in base 15 = aaa and consists of only the digit 'a'.
2410 in base 18 = 77g and consists of only the digits '7' and 'g'.
2410 in base 24 = 44a and consists of only the digits '4' and 'a'.
2410 in base 28 = 322 and consists of only the digits '2' and '3'.
2410 in base 34 = 22u and consists of only the digits '2' and 'u'.

The number 2410 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000094: Number of trees of diameter 4.
A025591: Maximal coefficient of Product_{k<=n} (x^k+1). Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0 or 1.
A047966: a(n) = Sum_{ d divides n } q(d), where q(d) = A000009 = number of partitions of d into distinct parts.
A063866: Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 1.
A102726: Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.
A121030: Multiples of 10 containing a 10 in their decimal representation.
A179888: Starting with a(1)=2: if m is a term then also 4*m+1 and 4*m+2.
A220010: T(n,k)=Number of nXk arrays of the minimum value of corresponding elements and their horizontal, vertical or antidiagonal neighbors in a random, but sorted with lexicographically nondecreasing rows and nonincreasing columns, 0..3 nXk array
A242994: Numbers n such that n!3 - 3 is prime, where n!3 = n!!! is a triple factorial number (A007661).
A283784: T(n,k)=Number of nXk 0..1 arrays with no 1 equal to more than two of its horizontal, diagonal and antidiagonal neighbors, with the exception of exactly two elements.

Number of the day: 768721

Properties of the number 768721:

768721 is a cyclic number.
768721 = 19 × 40459 is semiprime and squarefree.
768721 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 728244 totatives.
768721 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 768721 results in a prime.
768721 = 384361² - 384360² = 20239² - 20220² is the difference of 2 nonnegative squares in 2 ways.
768721 is the difference of 2 positive pentagonal numbers in 2 ways.
768721 = 61² + 90² + 870² is the sum of 3 positive squares.
768721² is the sum of 3 positive squares.
768721 is a proper divisor of 1861⁶⁶ - 1.
768721 is an emirpimes in (at least) the following bases: 3, 4, 5, 9, 17, 19, 28, 33, 34, 35, 41, 43, 45, 51, 52, 53, 55, 61, 62, 65, 69, 71, 73, 74, 75, 79, 84, 89, 90, 91, 93, and 97.
768721 in base 3 = 1110001111011 and consists of only the digits '0' and '1'.

The number 768721 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A245157: G.f.: Sum_{n>=0} x^n/((1+x)^(2*n+1)*(1 - (2*n+1)*x)).

Number of the day: 897778

Properties of the number 897778:

897778 = 2 × 7² × 9161 is the 826654^th composite number and is not squarefree.
897778 has 3 distinct prime factors, 12 divisors, 29 antidivisors and 384720 totatives.
897778 has a semiprime digit sum 46 in base 10.
Reversing the decimal digits of 897778 results in a semiprime.
897778 is the sum of 2 positive triangular numbers.
897778 is the difference of 2 positive pentagonal numbers in 6 ways.
897778 = 287² + 903² is the sum of 2 positive squares in 1 way.
897778 = 51² + 116² + 939² is the sum of 3 positive squares.
897778² = 518322² + 733040² is the sum of 2 positive squares in 1 way.
897778² is the sum of 3 positive squares.
897778 is a proper divisor of 5⁴²⁰ - 1.

Number of the day: 19843

Properties of the number 19843:

19843 is a cyclic number.
19841 and 19843 form a twin prime pair.
19843 has 5 antidivisors and 19842 totatives.
19843 has a semiprime digit sum 25 in base 10.
Reversing the decimal digits of 19843 results in a sphenic number.
19843 = 9922² - 9921² is the difference of 2 nonnegative squares in 1 way.
19843 is the difference of 2 positive pentagonal numbers in 1 way.
19843 = 23² + 33² + 135² is the sum of 3 positive squares.
19843² is the sum of 3 positive squares.
19843 is a proper divisor of 7³³⁰⁷ - 1.
19843 is an emirp in (at least) the following bases: 2, 7, 12, 13, 21, 23, 37, 47, 50, 51, 55, 62, 65, 76, 77, 91, 93, 95, 97, and 99.
19843 is palindromic in (at least) the following bases: 80, -47, and -57.
19843 in base 5 = 1113333 and consists of only the digits '1' and '3'.
19843 in base 31 = kk3 and consists of only the digits '3' and 'k'.
19843 in base 44 = AAh and consists of only the digits 'A' and 'h'.
19843 in base 46 = 9HH and consists of only the digits '9' and 'H'.
19843 in base 57 = 667 and consists of only the digits '6' and '7'.

The number 19843 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A061331: Primes with 19 as smallest positive primitive root.
A078746: a(n) = prime(2*n*(n+1)+1).
A082079: Balanced primes of order four.
A103384: Primes in A103374.
A119596: Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.
A127990: Largest prime factor of 2*n^3 - 2*n + 9.
A154624: Primes congruent to 34 mod 71.
A162307: Primes of the form k*(k+2)/3 - 2, k > 0.
A292578: Primes of the form 11*n^2 + 55*n + 43.
A298560: T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 5 or 7 king-move adjacent elements, with upper left element zero.

Number of the day: 5826

Constantin Carathéodory was born on this day 147 years ago.

Properties of the number 5826:

5826 is the 1454^th totient number.
5826 = 2 × 3 × 971 is a sphenic number and squarefree.
5826 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 1940 totatives.
5826 has a semiprime digit sum 21 in base 10.
5826 has a Fibonacci digit sum 21 in base 10.
5826 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 5826 results in a sphenic number.
5826 is the difference of 2 positive pentagonal numbers in 2 ways.
5826 = 1² + 40² + 65² is the sum of 3 positive squares.
5826² is the sum of 3 positive squares.
5826 is a proper divisor of 239¹⁰ - 1.
5826 = '58' + '26' is the concatenation of 2 emirpimes.
5826 is palindromic in (at least) the following bases: 30, 52, -41, -56, and -64.
5826 in base 7 = 22662 and consists of only the digits '2' and '6'.
5826 in base 18 = hhc and consists of only the digits 'c' and 'h'.
5826 in base 29 = 6qq and consists of only the digits '6' and 'q'.
5826 in base 30 = 6e6 and consists of only the digits '6' and 'e'.
5826 in base 51 = 2CC and consists of only the digits '2' and 'C'.
5826 in base 52 = 282 and consists of only the digits '2' and '8'.

The number 5826 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A015863: Numbers n such that sigma(n) = sigma(n + 4).
A054352: Lengths of successive generations of the Kolakoski sequence A000002.
A059468: Numbers which are the sum of their proper divisors containing the digit 9.
A079232: Number of non-associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.
A125221: Numbers k such that binomial(3k, k) + 1 is prime.
A234800: First occurrence of n in A234323: Number of nontrivial zeros of the Riemann Zeta function in the interval 1/2+i[n,n+1).
A234804: Positions of 3's in A234323.
A261942: T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each row nonprime and column prime, read as a binary number with top and left being the most significant bits.
A266886: T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with no element 1 greater than its north, northeast or northwest neighbor modulo n and the upper left element equal to 0.
A284119: Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

Number of the day: 4637

Properties of the number 4637:

4637 is a cyclic number.
4637 and 4639 form a twin prime pair.
4637 has 17 antidivisors and 4636 totatives.
4637 has an oblong digit sum 20 in base 10.
4637 has sum of divisors equal to 4638 which is a sphenic number.
4637 = 2319² - 2318² is the difference of 2 nonnegative squares in 1 way.
4637 is the sum of 2 positive triangular numbers.
4637 is the difference of 2 positive pentagonal numbers in 1 way.
4637 = 34² + 59² is the sum of 2 positive squares in 1 way.
4637 = 2² + 3² + 68² is the sum of 3 positive squares.
4637² = 2325² + 4012² is the sum of 2 positive squares in 1 way.
4637² is the sum of 3 positive squares.
4637 is a proper divisor of 173³⁸ - 1.
4637 = '463' + '7' is the concatenation of 2 prime numbers.
4637 is an emirp in (at least) the following bases: 2, 3, 4, 5, 19, 21, 22, 25, 28, 29, 31, 35, 47, 52, 54, 58, 59, 64, 66, 67, 69, 71, 73, 76, 78, 80, 84, 85, 89, 93, 94, and 99.
4637 is palindromic in (at least) the following bases: 45, 61, and -76.
4637 in base 20 = bbh and consists of only the digits 'b' and 'h'.
4637 in base 21 = aah and consists of only the digits 'a' and 'h'.
4637 in base 44 = 2HH and consists of only the digits '2' and 'H'.
4637 in base 45 = 2D2 and consists of only the digits '2' and 'D'.
4637 in base 60 = 1HH and consists of only the digits '1' and 'H'.
4637 in base 61 = 1F1 and consists of only the digits '1' and 'F'.

The number 4637 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001428: Number of inverse semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).
A003378: Numbers that are the sum of 11 positive 7th powers.
A007641: Primes of the form 2*n^2 + 29.
A011757: a(n) = prime(n^2).
A022004: Initial members of prime triples (p, p+2, p+6).
A025147: Number of partitions of n into distinct parts >= 2.
A078847: 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 pattern = [2, 4, 6]; short notation = [246] d-pattern.
A078946: Primes p such that p, p+2, p+6, p+12 and p+14 are consecutive primes.
A146360: Primes p such that continued fraction of (1 + sqrt(p))/2 has period 15: primes in A146338.
A236458: Primes p with p + 2 and prime(p) + 2 both prime.

Number of the day: 1521

Properties of the number 1521:

1521 = 3² × 13² is the 1280^th composite number and is not squarefree.
1521 has 2 distinct prime factors, 9 divisors, 10 antidivisors and 936 totatives.
1521 = 39² is a perfect power of an emirpimes.
1521 has a semiprime digit sum 9 in base 10.
1521 has a semiprime digit product 10 in base 10.
1521 has a triangular digit product 10 in base 10.
1521 has sum of divisors equal to 2379 which is a sphenic number.
1521 = 39² is a perfect square.
1521 = (38 × 39)/2 + (39 × 40)/2 is the sum of at least 2 consecutive triangular numbers in 1 way. In fact, it is the sum of 2 triangular numbers.
1521 = 761² - 760² = 255² - 252² = 89² - 80² = 65² - 52² = 39² - 0² is the difference of 2 nonnegative squares in 5 ways.
1521 is the difference of 2 positive pentagonal numbers in 1 way.
1521 = 15² + 36² is the sum of 2 positive squares in 1 way.
1521 is the sum of 3 positive squares.
1521² = 585² + 1404² = 1071² + 1080² is the sum of 2 positive squares in 2 ways.
1521² is the sum of 3 positive squares.
1521 is a proper divisor of 991³ - 1.
1521 = '15' + '21' is the concatenation of 2 semiprime numbers.
1521 = '15' + '21' is the concatenation of 2 triangular numbers.
1521 is palindromic in (at least) the following bases: 15, 22, 38, -23, -31, -40, -76, -80, and -95.
1521 in base 13 = 900 and consists of only the digits '0' and '9'.
1521 in base 15 = 6b6 and consists of only the digits '6' and 'b'.
1521 in base 19 = 441 and consists of only the digits '1' and '4'.
1521 in base 21 = 399 and consists of only the digits '3' and '9'.
1521 in base 22 = 333 and consists of only the digit '3'.
1521 in base 27 = 229 and consists of only the digits '2' and '9'.
1521 in base 37 = 144 and consists of only the digits '1' and '4'.
1521 in base 38 = 121 and consists of only the digits '1' and '2'.
1521 in base 39 = 100 and consists of only the digits '0' and '1'.

The number 1521 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000290: The squares: a(n) = n^2.
A001597: Perfect powers: m^k where m > 0 and k >= 2.
A003238: Number of rooted trees with n vertices in which vertices at the same level have the same degree.
A007955: Product of divisors of n.
A016754: Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.
A057660: a(n) = Sum_{k=1..n} n/gcd(n,k).
A080335: Diagonal in square spiral or maze arrangement of natural numbers.
A085986: Squares of the squarefree semiprimes (p^2*q^2).
A195160: Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...
A299287: Coordination sequence for "tcd" 3D uniform tiling.

Number of the day: 690827

Properties of the number 690827:

690827 is a cyclic number.
690827 = 37 × 18671 is semiprime and squarefree.
690827 has 2 distinct prime factors, 4 divisors, 43 antidivisors and 672120 totatives.
690827 = 345414² - 345413² = 9354² - 9317² is the difference of 2 nonnegative squares in 2 ways.
690827 is the difference of 2 positive pentagonal numbers in 2 ways.
690827 = 3² + 83² + 827² is the sum of 3 positive squares.
690827² = 224052² + 653485² is the sum of 2 positive squares in 1 way.
690827² is the sum of 3 positive squares.
690827 is a proper divisor of 593¹⁸⁶⁷ - 1.
690827 is an emirpimes in (at least) the following bases: 2, 7, 9, 11, 13, 14, 19, 24, 25, 28, 29, 31, 32, 35, 43, 46, 47, 49, 58, 64, 65, 66, 67, 69, 73, 75, 78, 81, 84, 85, 87, 89, 90, 91, 92, 93, 95, 97, 99, and 100.
690827 is palindromic in (at least) base -16.
690827 in base 30 = phhh and consists of only the digits 'h' and 'p'.

Number of the day: 922647

Marin Mersenne was born on this day 432 years ago.

Properties of the number 922647:

922647 = 3 × 11 × 73 × 383 is the 849714^th composite number and is squarefree.
922647 has 4 distinct prime factors, 16 divisors, 21 antidivisors and 550080 totatives.
922647 has a sphenic digit sum 30 in base 10.
922647 has an oblong digit sum 30 in base 10.
Reversing the decimal digits of 922647 results in a sphenic number.
922647 = 461324² - 461323² = 153776² - 153773² = 41944² - 41933² = 13996² - 13963² = 6356² - 6283² = 2216² - 1997² = 1396² - 1013² = 976² - 173² is the difference of 2 nonnegative squares in 8 ways.
922647 is the difference of 2 positive pentagonal numbers in 3 ways.
922647 is not the sum of 3 positive squares.
922647² = 606672² + 695145² is the sum of 2 positive squares in 1 way.
922647² is the sum of 3 positive squares.
922647 is a proper divisor of 1531⁹⁰ - 1.
922647 = '9' + '22647' is the concatenation of 2 semiprime numbers.

Number of the day: 114735

Properties of the number 114735:

114735 is a cyclic number.
114735 = 3 × 5 × 7649 is a sphenic number and squarefree.
114735 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 61184 totatives.
114735 has a semiprime digit sum 21 in base 10.
114735 has a Fibonacci digit sum 21 in base 10.
114735 has a triangular digit sum 21 in base 10.
114735 has an oblong digit product 420 in base 10.
114735 = 57368² - 57367² = 19124² - 19121² = 11476² - 11471² = 3832² - 3817² is the difference of 2 nonnegative squares in 4 ways.
114735 is the difference of 2 positive pentagonal numbers in 2 ways.
114735 is not the sum of 3 positive squares.
114735² = 68841² + 91788² = 48132² + 104151² = 75369² + 86508² = 23985² + 112200² is the sum of 2 positive squares in 4 ways.
114735² is the sum of 3 positive squares.
114735 is a proper divisor of 331²³⁹ - 1.
114735 = '1147' + '35' is the concatenation of 2 semiprime numbers.
114735 is palindromic in (at least) the following bases: 54, 82, 89, and -66.
114735 in base 54 = dId and consists of only the digits 'I' and 'd'.

The number 114735 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A134186: A 3 person 9 X 9 Markov approach to a zero sum game where: Sum[game_value(MAi),{i,1,3}]=0 and two of the games are minimal Pisot vector Markovs and the third is a negative Fibonacci: Characteristic Polynomial: -1 + 3 x^2 + 3 x^3 - 4 x^4 - 5 x^5 + x^6 + 4 x^7 - x^9; MA1={{0,1,0},{,0,0,1},{1,1,0}};Det=1 ;gv=-1/4; MA2={{0,1,1},{1,0,0},{0,1,0}};Det=1;gv=-1/4 MA2={{0,0,1},{0,1,0},{1,0,-1}};Det=-1;gv=1/2.
A140227: Binomial transform of [1, 4, 6, 4, 1, 1, -1, 1, -1, 1,...].

Number of the day: 46585

Elwyn Ralph Berlekamp was born on this day 80 years ago.

Properties of the number 46585:

46585 = 5 × 7 × 11³ is the 41771^th composite number and is not squarefree.
46585 has 3 distinct prime factors, 16 divisors, 21 antidivisors and 29040 totatives.
46585 has a triangular digit sum 28 in base 10.
46585 = 60² + … + 70² is the sum of at least 2 consecutive positive squares in 1 way.
46585 = 33³ + 22³ is the sum of 2 positive cubes in 1 way.
46585 = (134 × 135)/2 + … + (138 × 139)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
46585 = 23293² - 23292² = 4661² - 4656² = 3331² - 3324² = 2123² - 2112² = 683² - 648² = 451² - 396² = 341² - 264² = 253² - 132² is the difference of 2 nonnegative squares in 8 ways.
46585 is the difference of 2 positive pentagonal numbers in 8 ways.
46585 = 37² + 60² + 204² is the sum of 3 positive squares.
46585² = 27951² + 37268² is the sum of 2 positive squares in 1 way.
46585² is the sum of 3 positive squares.
46585 is a proper divisor of 727⁴⁴ - 1.
46585 is palindromic in (at least) the following bases: 71, and -69.
46585 in base 49 = JJZ and consists of only the digits 'J' and 'Z'.

The number 46585 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A013618: Triangle of coefficients in expansion of (1+11x)^n.
A033445: a(n) = (n - 1)*(n^2 + n - 1).
A038315: Triangle whose (i,j)-th entry is binomial(i,j)*11^(i-j)*1^j.
A057965: Triangle T(n,k) of number of minimal 4-covers of a labeled n-set that cover k points of that set uniquely (k=4,..,n).
A120075: Row sums of triangle A120073 (denominator triangle for H atom spectrum).
A192100: Table read by rows of numbers of unordered pairs of partitions of n-element set that have Rand distance k (n>=2, 1 <= k <= n(n-1)/2.
A261640: Numbers n such that both the digital sum of n is the same as the digital sum of n^2 in both base 2 and base 10.
A267233: Number of length-4 0..n arrays with no following elements greater than or equal to the first repeated value.
A325179: Heinz numbers of integer partitions such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is 1.

Number of the day: 140218

Properties of the number 140218:

140218 = 2 × 13 × 5393 is a sphenic number and squarefree.
140218 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 64704 totatives.
Reversing the decimal digits of 140218 results in a sphenic number.
140218 is the sum of 2 positive triangular numbers.
140218 is the difference of 2 positive pentagonal numbers in 4 ways.
140218 = 113² + 357² = 33² + 373² is the sum of 2 positive squares in 2 ways.
140218 = 24² + 59² + 369² is the sum of 3 positive squares.
140218² = 80682² + 114680² = 53930² + 129432² = 24618² + 138040² = 30368² + 136890² is the sum of 2 positive squares in 4 ways.
140218² is the sum of 3 positive squares.
140218 is a proper divisor of 53³³⁷ - 1.
140218 is palindromic in (at least) base 62.
140218 in base 24 = a3aa and consists of only the digits '3' and 'a'.
140218 in base 62 = aTa and consists of only the digits 'T' and 'a'.

Number of the day: 95101

Properties of the number 95101:

95101 is a cyclic number.
95101 is the 9168^th prime.
95101 has 9 antidivisors and 95100 totatives.
95101 has sum of divisors equal to 95102 which is a sphenic number.
Reversing the decimal digits of 95101 results in an emirp.
95101 = 47551² - 47550² is the difference of 2 nonnegative squares in 1 way.
95101 is the sum of 2 positive triangular numbers.
95101 is the difference of 2 positive pentagonal numbers in 1 way.
95101 = 149² + 270² is the sum of 2 positive squares in 1 way.
95101 = 50² + 51² + 300² is the sum of 3 positive squares.
95101² = 50699² + 80460² is the sum of 2 positive squares in 1 way.
95101² is the sum of 3 positive squares.
95101 is a proper divisor of 733³⁰⁰ - 1.
95101 is an emirp in (at least) the following bases: 9, 10, 11, 15, 21, 23, 25, 26, 29, 35, 42, 43, 45, 49, 52, 55, 59, 67, 70, 85, and 92.
95101 is palindromic in (at least) the following bases: 69, 84, and -51.
95101 in base 48 = fDD and consists of only the digits 'D' and 'f'.

The number 95101 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A032629: Primes that are concatenations of n with n + 6.
A105010: Primes from merging of 5 successive digits in decimal expansion of exp(Pi).
A154822: Primes p of the form : p+p^2+p^3-+4=prime.
A167626: Prime numbers ending in the prime number 101.
A179037: Emirps with a 5 as the only prime digit.
A183546: Number of n X n 0..2 arrays with each element equal to either the maximum or the minimum of its horizontal and vertical neighbors
A183549: Number of nX4 0..2 arrays with each element equal to either the maximum or the minimum of its horizontal and vertical neighbors
A183554: T(n,k)=Number of nXk 0..2 arrays with each element equal to either the maximum or the minimum of its horizontal and vertical neighbors

Number of the day: 766486

James Joseph Sylvester was born on this day 206 years ago.

Solomon Lefschetz was born on this day 136 years ago.

Properties of the number 766486:

766486 = 2 × 7 × 53 × 1033 is the 705009^th composite number and is squarefree.
766486 has 4 distinct prime factors, 16 divisors, 23 antidivisors and 321984 totatives.
766486 has an emirp digit sum 37 in base 10.
Reversing the decimal digits of 766486 results in a sphenic number.
766486 is the sum of 2 positive triangular numbers.
766486 is the difference of 2 positive pentagonal numbers in 4 ways.
766486 = 1² + 66² + 873² is the sum of 3 positive squares.
766486² = 518840² + 564186² = 404936² + 650790² = 276920² + 714714² = 142464² + 753130² is the sum of 2 positive squares in 4 ways.
766486² is the sum of 3 positive squares.
766486 is a proper divisor of 1697¹² - 1.

Number of the day: 5962645244

Properties of the number 5962645244:

5962645244 = 2² × 1490661311 is the 5684760115^th composite number and is not squarefree.
5962645244 has 2 distinct prime factors, 6 divisors, 17 antidivisors and 2981322620 totatives.
5962645244 has a prime digit sum 47 in base 10.
Reversing the decimal digits of 5962645244 results in a sphenic number.
5962645244 = 1490661312² - 1490661310² is the difference of 2 nonnegative squares in 1 way.
5962645244 is the difference of 2 positive pentagonal numbers in 1 way.
5962645244 is not the sum of 3 positive squares.
5962645244² is the sum of 3 positive squares.
5962645244 is a proper divisor of 13^149066131 - 1.

Number of the day: 792285

Properties of the number 792285:

792285 = 3 × 5 × 13 × 17 × 239 is the 728904^th composite number and is squarefree.
792285 has 5 distinct prime factors, 32 divisors, 33 antidivisors and 365568 totatives.
792285 has a semiprime digit sum 33 in base 10.
Reversing the decimal digits of 792285 results in a sphenic number.
792285 is the difference of 2 nonnegative squares in 16 ways.
792285 is the difference of 2 positive pentagonal numbers in 8 ways.
792285 = 76² + 125² + 878² is the sum of 3 positive squares.
792285² = 475371² + 633828² = 335556² + 717717² = 121173² + 782964² = 533448² + 585789² = 195024² + 767907² = 189288² + 769341² = 412992² + 676131² = 402237² + 682584² = 33699² + 791568² = 501900² + 613035² = 304725² + 731340² = 75285² + 788700² = 372840² + 699075² is the sum of 2 positive squares in 13 ways.
792285² is the sum of 3 positive squares.
792285 is a proper divisor of 1913²⁴ - 1.

The number 792285 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A089075: A theta1 minimal Pisot type valued matrix 4 X 4 4-Bonacci matrix from a root of polynomial of: x^4-x^3-1=0.