Number of the day: 8346

Properties of the number 8346:

8346 = 2 × 3 × 13 × 107 is the 7300^th composite number and is squarefree.
8346 has 4 distinct prime factors, 16 divisors, 7 antidivisors and 2544 totatives.
8346 has a semiprime digit sum 21 in base 10.
8346 has a Fibonacci digit sum 21 in base 10.
8346 has a triangular digit sum 21 in base 10.
8346 is the difference of 2 positive pentagonal numbers in 3 ways.
8346 = 4² + 49² + 77² is the sum of 3 positive squares.
8346² = 3210² + 7704² is the sum of 2 positive squares in 1 way.
8346² is the sum of 3 positive squares.
8346 is a proper divisor of 857² - 1.
8346 is palindromic in (at least) the following bases: 43, and 56.
8346 in base 7 = 33222 and consists of only the digits '2' and '3'.
8346 in base 34 = 77g and consists of only the digits '7' and 'g'.
8346 in base 42 = 4UU and consists of only the digits '4' and 'U'.
8346 in base 43 = 4M4 and consists of only the digits '4' and 'M'.
8346 in base 55 = 2ff and consists of only the digits '2' and 'f'.
8346 in base 56 = 2b2 and consists of only the digits '2' and 'b'.

The number 8346 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A022268: a(n) = n*(11*n - 1)/2.
A041746: Numerators of continued fraction convergents to sqrt(393).
A129025: See Mathematica code.
A213271: Costas arrays such that the corresponding permutation is a derangement.
A216423: Numbers k such that 4^k + k^4 + 1 is prime.
A217390: Numbers n such that sum of squares of digits of n equals the sum of prime divisors of n.
A252407: T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and diagonal sum equal to 2 3 4 6 or 7 and every 3X3 column and antidiagonal sum not equal to 2 3 4 6 or 7
A269967: Indices of zeros in A269783.
A330791: Values of k such that A005132(k) (the k-th number in the Recamn sequence) divides k.
A341477: Coefficients related to the asymptotics of generalized Delannoy numbers.

Number of the day: 4252

Properties of the number 4252:

4252 is the 1081^th totient number.
4252 = 2² × 1063 is the 3669^th composite number and is not squarefree.
4252 has 2 distinct prime factors, 6 divisors, 25 antidivisors and 2124 totatives.
4252 has an emirp digit sum 13 in base 10.
4252 has a Fibonacci digit sum 13 in base 10.
4252 = 1064² - 1062² is the difference of 2 nonnegative squares in 1 way.
4252 is the sum of 2 positive triangular numbers.
4252 is the difference of 2 positive pentagonal numbers in 1 way.
4252 is not the sum of 3 positive squares.
4252² is the sum of 3 positive squares.
4252 is a proper divisor of 719⁶ - 1.
4252 is palindromic in (at least) the following bases: -13, -21, -31, -36, and -50.
4252 in base 3 = 12211111 and consists of only the digits '1' and '2'.
4252 in base 20 = acc and consists of only the digits 'a' and 'c'.
4252 in base 32 = 44s and consists of only the digits '4' and 's'.
4252 in base 37 = 33Y and consists of only the digits '3' and 'Y'.

The number 4252 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A055328: Number of rooted identity trees with n nodes and 3 leaves.
A077297: Partition the concatenation 1234567...of natural numbers into successive strings which are multiples of 4, all different and > 4. ( 0's never taken as the most significant digit.)
A090748: Numbers n such that 2^(n+1) - 1 is prime.
A134212: Positions of 12 after decimal point in decimal expansion of Pi.
A193492: Put the natural numbers together without spaces and read them four at a time advancing one space each time.
A212585: Walks of length n on the x-axis using steps {1,-1} and visiting no point more than twice.
A239262: Number of partitions of n having (sum of odd parts) > (sum of even parts).
A272790: Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 537", based on the 5-celled von Neumann neighborhood.
A320234: Expansion of Product_{k=1..8} theta_3(q^k), where theta_3() is the Jacobi theta function.
A353255: Expansion of Sum_{k>=0} x^k * Product_{j=0..k-1} (2 * j + x).

Number of the day: 7113

Properties of the number 7113:

7113 = 3 × 2371 is semiprime and squarefree.
7113 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 4740 totatives.
7113 has an oblong digit sum 12 in base 10.
7113 has a semiprime digit product 21 in base 10.
7113 has a Fibonacci digit product 21 in base 10.
7113 has a triangular digit product 21 in base 10.
Reversing the decimal digits of 7113 results in an emirpimes.
7113 = 3557² - 3556² = 1187² - 1184² is the difference of 2 nonnegative squares in 2 ways.
7113 is the sum of 2 positive triangular numbers.
7113 is the difference of 2 positive pentagonal numbers in 1 way.
7113 = 10² + 17² + 82² is the sum of 3 positive squares.
7113² is the sum of 3 positive squares.
7113 is a proper divisor of 1907⁶ - 1.
7113 = '7' + '113' is the concatenation of 2 prime numbers.
7113 is an emirpimes in (at least) the following bases: 4, 5, 8, 9, 10, 19, 28, 36, 44, 47, 48, 51, 60, 64, 65, 69, 70, 71, 72, 73, 75, 76, 84, 89, 94, and 100.
7113 is palindromic in (at least) the following bases: 29, 45, and -34.
7113 in base 29 = 8d8 and consists of only the digits '8' and 'd'.
7113 in base 44 = 3TT and consists of only the digits '3' and 'T'.
7113 in base 45 = 3N3 and consists of only the digits '3' and 'N'.
7113 in base 59 = 22X and consists of only the digits '2' and 'X'.

The number 7113 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006124: a(n) = 3 + n/2 + 7*n^2/2.
A022503: Describe the previous term! (method B - initial term is 7).
A027026: a(n) = T(n,n+4), T given by A027023.
A027032: a(n) = T(n,2n-8), T given by A027023.
A033679: a(1) = 2; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A152136: a(0) = 0; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any two consecutive digits in the sequence is a prime.
A152607: a(1) = 1; thereafter a(n) is always the smallest integer > a(n-1) not leading to a contradiction, such that the concatenation of any two consecutive digits in the sequence is a prime.
A239986: T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or two plus the sum of the elements above it or one plus the sum of the elements diagonally to its northwest, modulo 4
A249334: Numbers n for which the digital sum contains the same distinct digits as the digital product.
A249335: Numbers n for which the digital sum contains the same distinct digits as the digital product but the digital sum is not equal to the digital product.

Number of the day: 11168

Johann Bernoulli was born on this day 355 years ago.

Properties of the number 11168:

11168 = 2⁵ × 349 is the 9815^th composite number and is not squarefree.
11168 has 2 distinct prime factors, 12 divisors, 9 antidivisors and 5568 totatives.
11168 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 11168 results in a prime.
11168 = 2³ + 8³ + 22³ is the sum of 3 positive cubes in 1 way.
11168 = 2793² - 2791² = 1398² - 1394² = 702² - 694² = 357² - 341² is the difference of 2 nonnegative squares in 4 ways.
11168 = 52² + 92² is the sum of 2 positive squares in 1 way.
11168 = 12² + 32² + 100² is the sum of 3 positive squares.
11168² = 5760² + 9568² is the sum of 2 positive squares in 1 way.
11168² is the sum of 3 positive squares.
11168 is a proper divisor of 911⁴ - 1.
11168 is palindromic in (at least) the following bases: 36, 55, and -25.
11168 in base 13 = 5111 and consists of only the digits '1' and '5'.
11168 in base 22 = 111e and consists of only the digits '1' and 'e'.
11168 in base 30 = cc8 and consists of only the digits '8' and 'c'.
11168 in base 36 = 8m8 and consists of only the digits '8' and 'm'.
11168 in base 54 = 3ii and consists of only the digits '3' and 'i'.
11168 in base 55 = 3c3 and consists of only the digits '3' and 'c'.

The number 11168 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A046994: Number of Greek-key tours on a 3 X n board; i.e., self-avoiding walks on a 3 X n grid starting in the top left corner.
A084422: Number of subsets of integers 1 through n (including the empty set) containing no pair of integers that share a common factor.
A110735: Let n = a_1a_2...a_k, where the a_i are digits. a(n) = least multiple of n of the type b_1a_1b_2a_2...a_kb_{k+1}, obtained by inserting single digits b_i in the gaps and both ends; 0 if no such number exists.
A138744: Let r_1 = 1. Let r_{m+1} = r_1 + 1/(r_2 + 1/(r_3 +...(r_{m-1} + 1/r_m)...)), a continued fraction of rational terms. Then a(n) is the sum of the (positive integer) terms in the simple continued fraction of r_n.
A164191: Number of binary strings of length n with equal numbers of 00000 and 11111 substrings
A177482: Suppose e<f, e<h, g<f and g<h. To avoid fegh means not to have four consecutive letters such that the second and the third letters are less than the first and the fourth letters.
A185098: a(n) = floor((265/6)*4^(n-4) - n^2 - ((15+(-1)^(n-1))/6)* 2^(n-3)).
A223432: T(n,k)=5X5X5 triangular graph without horizontal edges coloring a rectangular array: number of nXk 0..14 arrays where 0..14 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 3,6 3,7 4,7 4,8 5,8 5,9 6,10 6,11 7,11 7,12 8,12 8,13 9,13 9,14 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph
A248575: Rounded sums of the non-integer cube roots of n, as partitioned by the integer roots: round[sum(j from n^3+1 to (n+1)^3-1, j^(1/3))].
A285152: T(n,k) = Number of n X k 0..1 arrays with the number of 1s horizontally or antidiagonally adjacent to some 0 one less than the number of 0's adjacent to some 1.

Number of the day: 2613

Niels Henrik Abel was born on this day 220 years ago.

Properties of the number 2613:

2613 = 3 × 13 × 67 is a sphenic number and squarefree.
2613 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 1584 totatives.
2613 has an oblong digit sum 12 in base 10.
2613 has a triangular digit product 36 in base 10.
2613 = 1307² - 1306² = 437² - 434² = 107² - 94² = 53² - 14² is the difference of 2 nonnegative squares in 4 ways.
2613 is the sum of 2 positive triangular numbers.
2613 is the difference of 2 positive pentagonal numbers in 2 ways.
2613 = 7² + 8² + 50² is the sum of 3 positive squares.
2613² = 1005² + 2412² is the sum of 2 positive squares in 1 way.
2613² is the sum of 3 positive squares.
2613 is a proper divisor of 937² - 1.
2613 = '2' + '613' is the concatenation of 2 prime numbers.
2613 is palindromic in (at least) the following bases: 29, 66, and -30.
2613 in base 25 = 44d and consists of only the digits '4' and 'd'.
2613 in base 28 = 399 and consists of only the digits '3' and '9'.
2613 in base 29 = 333 and consists of only the digit '3'.

The number 2613 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A022288: a(n) = n*(31*n-1)/2.
A121033: Multiples of 13 containing a 13 in their decimal representation.
A130689: Number of partitions of n such that every part divides the largest part; a(0) = 1.
A140674: a(n) = n*(3*n + 17)/2.
A153875: 3 times 13-gonal (or tridecagonal) numbers: 3*n*(11*n - 9)/2.
A209032: T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and first differences in -k..k.
A231855: T(n,k)=Number of nXk 0..2 arrays with no element having a strict majority of its horizontal and antidiagonal neighbors equal to itself plus one mod 3, with upper left element zero (rock paper and scissors drawn positions)
A244788: T(n,k)=Number of length n 0..k arrays with each partial sum starting from the beginning no more than one standard deviation from its mean.
A349053: Number of non-weakly alternating integer compositions of n.
A350844: Number of strict integer partitions of n with no difference -2.

Number of the day: 48808

William Rowan Hamilton was born on this day 217 years ago.

John Venn was born on this day 188 years ago.

Properties of the number 48808:

48808 = 2³ × 6101 is the 43789^th composite number and is not squarefree.
48808 has 2 distinct prime factors, 8 divisors, 13 antidivisors and 24400 totatives.
48808 has a triangular digit sum 28 in base 10.
48808 = 12203² - 12201² = 6103² - 6099² is the difference of 2 nonnegative squares in 2 ways.
48808 is the sum of 2 positive triangular numbers.
48808 is the difference of 2 positive pentagonal numbers in 2 ways.
48808 = 98² + 198² is the sum of 2 positive squares in 1 way.
48808 = 8² + 150² + 162² is the sum of 3 positive squares.
48808² = 29600² + 38808² is the sum of 2 positive squares in 1 way.
48808² is the sum of 3 positive squares.
48808 is a proper divisor of 1723¹⁰ - 1.
48808 is palindromic in (at least) the following bases: 47, -57, -59, and -80.
48808 in base 47 = M4M and consists of only the digits '4' and 'M'.

The number 48808 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A175103: Integers n such that 17+30*n are terms in A172456.
A248488: Number of length n+5 0..7 arrays with some three disjoint pairs in each consecutive six terms having the same sum.
A248493: Number of length 4+5 0..n arrays with some three disjoint pairs in each consecutive six terms having the same sum

Number of the day: 124586

Properties of the number 124586:

124586 = 2 × 7 × 11 × 809 is the 112885^th composite number and is squarefree.
124586 has 4 distinct prime factors, 16 divisors, 15 antidivisors and 48480 totatives.
124586 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 124586 results in a semiprime.
124586 is the difference of 2 positive pentagonal numbers in 3 ways.
124586 = 9² + 64² + 347² is the sum of 3 positive squares.
124586² = 43120² + 116886² is the sum of 2 positive squares in 1 way.
124586² is the sum of 3 positive squares.
124586 is a proper divisor of 491²⁰ - 1.
124586 = '12458' + '6' is the concatenation of 2 semiprime numbers.
124586 is palindromic in (at least) the following bases: 52, 78, and -76.
124586 in base 3 = 20022220022 and consists of only the digits '0' and '2'.
124586 in base 52 = k3k and consists of only the digits '3' and 'k'.
124586 in base 58 = b22 and consists of only the digits '2' and 'b'.

The number 124586 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A351716: Starts of runs of 3 consecutive Lucas-Niven numbers (A351714).

Number of the day: 6971

Properties of the number 6971:

6971 is a cyclic number.
6971 is the 896^th prime.
6971 has 7 antidivisors and 6970 totatives.
6971 has a prime digit sum 23 in base 10.
6971 has a triangular digit product 378 in base 10.
6971 has sum of divisors equal to 6972 which is an oblong number.
6971 = 3486² - 3485² is the difference of 2 nonnegative squares in 1 way.
6971 is the difference of 2 positive pentagonal numbers in 1 way.
6971 = 1² + 9² + 83² is the sum of 3 positive squares.
6971² is the sum of 3 positive squares.
6971 is a proper divisor of 1279³⁴ - 1.
6971 is an emirp in (at least) the following bases: 4, 5, 7, 9, 13, 15, 19, 23, 29, 30, 31, 39, 41, 43, 53, 55, 57, 60, 61, 64, 69, 70, 73, 74, 75, 77, 79, 81, 83, 85, 89, 90, 91, 92, 97, and 99.
6971 is palindromic in (at least) the following bases: 82, -10, -35, -52, -69, and -85.
6971 in base 13 = 3233 and consists of only the digits '2' and '3'.
6971 in base 29 = 88b and consists of only the digits '8' and 'b'.
6971 in base 31 = 77r and consists of only the digits '7' and 'r'.
6971 in base 34 = 611 and consists of only the digits '1' and '6'.

The number 6971 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002148: Smallest prime p==3 (mod 8) such that Q(sqrt(-p)) has class number 2n+1.
A002327: Primes of the form k^2 - k - 1.
A057186: Numbers n such that (20^n+1)/21 is a prime.
A059236: Primes p such that x^41 = 2 has no solution mod p.
A082108: a(n) = 4n^2 + 6n + 1.
A089299: Number of square plane partitions of n.
A127341: Primes that can be written as the sum of 13 consecutive primes.
A136079: Father primes of order 10.
A141908: Primes congruent to 2 mod 23.
A153377: Larger of two consecutive prime numbers such that p1*p2*d + d = average of twin prime pairs, d (delta) = p2 - p1.

Number of the day: 2717

Otto Toeplitz was born on this day 141 years ago.

Properties of the number 2717:

2717 is a cyclic number.
2717 = 11 × 13 × 19 is a sphenic number and squarefree.
2717 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 2160 totatives.
2717 has an emirp digit sum 17 in base 10.
2717 = 2³ + 8³ + 13³ is the sum of 3 positive cubes in 1 way.
2717 = 14³ - 3³ is the difference of 2 positive cubes in 1 way.
2717 = 1359² - 1358² = 129² - 118² = 111² - 98² = 81² - 62² is the difference of 2 nonnegative squares in 4 ways.
2717 is the difference of 2 positive pentagonal numbers in 4 ways.
2717 = 2² + 3² + 52² is the sum of 3 positive squares.
2717² = 1045² + 2508² is the sum of 2 positive squares in 1 way.
2717² is the sum of 3 positive squares.
2717 is a proper divisor of 571² - 1.
2717 = '271' + '7' is the concatenation of 2 prime numbers.
2717 is palindromic in (at least) the following bases: -24, and -97.
2717 in base 23 = 533 and consists of only the digits '3' and '5'.

The number 2717 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000914: Stirling numbers of the first kind: s(n+2, n).
A033680: a(1) = 1; a(n) is smallest number >= a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A045947: Triangles in open triangular matchstick arrangement (triangle minus one side) of side n.
A054333: 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
A082985: Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
A111125: Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
A185589: Iterate the map in A006369 starting at 144.
A216169: Composite numbers > 9 which yield a prime whenever a 0 is inserted between any two digits.
A297720: T(n,k)=Number of nXk 0..1 arrays with every 1 horizontally, diagonally or antidiagonally adjacent to 1, 2 or 4 neighboring 1s.
A319721: Number of non-isomorphic antichains of multisets of weight n.

Number of the day: 629046

Properties of the number 629046:

629046 = 2 × 3⁴ × 11 × 353 is the 577774^th composite number and is not squarefree.
629046 has 4 distinct prime factors, 40 divisors, 23 antidivisors and 190080 totatives.
629046 = 1² + 14² + 793² is the sum of 3 positive squares.
629046² = 400950² + 484704² is the sum of 2 positive squares in 1 way.
629046² is the sum of 3 positive squares.
629046 is a proper divisor of 647¹⁶⁰ - 1.
629046 = '62' + '9046' is the concatenation of 2 semiprime numbers.

Number of the day: 202917

Properties of the number 202917:

202917 = 3 × 11² × 13 × 43 is the 184699^th composite number and is not squarefree.
202917 has 4 distinct prime factors, 24 divisors, 31 antidivisors and 110880 totatives.
202917 has a semiprime digit sum 21 in base 10.
202917 has a Fibonacci digit sum 21 in base 10.
202917 has a triangular digit sum 21 in base 10.
202917 is the difference of 2 nonnegative squares in 12 ways.
202917 is the sum of 2 positive triangular numbers.
202917 is the difference of 2 positive pentagonal numbers in 5 ways.
202917 = 2² + 47² + 448² is the sum of 3 positive squares.
202917² = 78045² + 187308² is the sum of 2 positive squares in 1 way.
202917² is the sum of 3 positive squares.
202917 is a proper divisor of 727¹⁴ - 1.
202917 = '2029' + '17' is the concatenation of 2 prime numbers.
202917 = '202' + '917' is the concatenation of 2 semiprime numbers.
202917 is palindromic in (at least) the following bases: -60, and -62.
202917 in base 32 = 6655 and consists of only the digits '5' and '6'.
202917 in base 34 = 55i5 and consists of only the digits '5' and 'i'.

The number 202917 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A218266: Number of standard Young tableaux of n cells and height >= 6.

Number of the day: 7793

Properties of the number 7793:

7793 is a cyclic number.
7793 is the 987^th prime.
7793 has 13 antidivisors and 7792 totatives.
7793 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 7793 results in a semiprime.
7793 = 3897² - 3896² is the difference of 2 nonnegative squares in 1 way.
7793 is the difference of 2 positive pentagonal numbers in 1 way.
7793 = 7² + 88² is the sum of 2 positive squares in 1 way.
7793 = 6² + 19² + 86² is the sum of 3 positive squares.
7793² = 1232² + 7695² is the sum of 2 positive squares in 1 way.
7793² is the sum of 3 positive squares.
7793 is a proper divisor of 43⁴⁸⁷ - 1.
7793 = '77' + '93' is the concatenation of 2 semiprime numbers.
7793 is an emirp in (at least) the following bases: 3, 4, 5, 11, 16, 20, 21, 23, 24, 27, 31, 41, 49, 55, 59, 61, 63, 64, 67, 70, 80, 83, 84, 85, 86, 95, 98, and 100.
7793 is palindromic in (at least) the following bases: 28, 53, -34, and -44.
7793 in base 28 = 9q9 and consists of only the digits '9' and 'q'.
7793 in base 33 = 755 and consists of only the digits '5' and '7'.
7793 in base 52 = 2jj and consists of only the digits '2' and 'j'.
7793 in base 53 = 2f2 and consists of only the digits '2' and 'f'.

The number 7793 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001134: Primes p such that the multiplicative order of 2 modulo p is (p-1)/4.
A051026: Number of primitive subsequences of {1, 2, ..., n}.
A057622: Initial prime in first sequence of n consecutive primes congruent to 5 modulo 6.
A068652: Numbers such that every cyclic permutation is a prime.
A128337: Numbers k such that (7^k + 5^k)/12 is prime.
A142200: Primes congruent to 3 mod 41.
A154577: Primes of the form 2n^2+14n+5.
A293663: Circular primes that are not repunits.
A316652: Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
A317716: Square array A(n, k), read by antidiagonals downwards: k-th prime p such that cyclic digit shifts produce exactly n different primes.

Number of the day: 75335

Ernst Friedrich Ferdinand Zermelo was born on this day 151 years ago.

Properties of the number 75335:

75335 = 5 × 13 × 19 × 61 is the 67912^th composite number and is squarefree.
75335 has 4 distinct prime factors, 16 divisors, 21 antidivisors and 51840 totatives.
75335 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 75335 results in a semiprime.
75335 = 37668² - 37667² = 7536² - 7531² = 2904² - 2891² = 1992² - 1973² = 648² - 587² = 612² - 547² = 444² - 349² = 276² - 29² is the difference of 2 nonnegative squares in 8 ways.
75335 is the difference of 2 positive pentagonal numbers in 7 ways.
75335 is not the sum of 3 positive squares.
75335² = 45201² + 60268² = 33592² + 67431² = 51129² + 55328² = 18544² + 73017² = 5073² + 75164² = 31407² + 68476² = 38247² + 64904² = 25916² + 70737² = 49324² + 56943² = 28975² + 69540² = 15960² + 73625² = 41040² + 63175² = 13585² + 74100² is the sum of 2 positive squares in 13 ways.
75335² is the sum of 3 positive squares.
75335 is a proper divisor of 1559¹⁰ - 1.
75335 = '753' + '35' is the concatenation of 2 semiprime numbers.
75335 is palindromic in (at least) base 70.
75335 in base 58 = MMp and consists of only the digits 'M' and 'p'.

The number 75335 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A132124: a(n) = n*(n+1)*(8*n + 1)/6.
A137080: Numbers k such that k and k^2 use only the digits 2, 3, 5, 6 and 7.
A345705: Numbers k such that (3^ord(3/2, k) - 2^ord(3/2, k))/k is a prime, where ord(3/2, k) is the multiplicative order of 3/2 (mod k).

Number of the day: 3430

Properties of the number 3430:

3430 = 2 × 5 × 7³ is the 2949^th composite number and is not squarefree.
3430 has 3 distinct prime factors, 16 divisors, 11 antidivisors and 1176 totatives.
3430 has a semiprime digit sum 10 in base 10.
3430 has a triangular digit sum 10 in base 10.
3430 = 7³ + 7³ + 14³ is the sum of 3 positive cubes in 1 way.
3430 is the sum of 2 positive triangular numbers.
3430 is the difference of 2 positive pentagonal numbers in 4 ways.
3430 = 9² + 10² + 57² is the sum of 3 positive squares.
3430² = 2058² + 2744² is the sum of 2 positive squares in 1 way.
3430² is the sum of 3 positive squares.
3430 is a proper divisor of 1373⁴ - 1.
3430 is palindromic in (at least) the following bases: 18, 21, 25, 69, 97, and -19.
3430 in base 16 = d66 and consists of only the digits '6' and 'd'.
3430 in base 18 = aaa and consists of only the digit 'a'.
3430 in base 19 = 99a and consists of only the digits '9' and 'a'.
3430 in base 21 = 7g7 and consists of only the digits '7' and 'g'.
3430 in base 24 = 5mm and consists of only the digits '5' and 'm'.
3430 in base 25 = 5c5 and consists of only the digits '5' and 'c'.
3430 in base 58 = 118 and consists of only the digits '1' and '8'.

The number 3430 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A007727: Number of 2n-bead black-white strings with n black beads and fundamental period 2n.
A025632: Numbers of form 7^i*10^j, with i, j >= 0.
A027466: Triangle whose (i,j)-th entry is binomial(i,j)*7^(i-j).
A027474: a(n) = 7^(n-2) * C(n,2).
A051682: 11-gonal (or hendecagonal) numbers: a(n) = n*(9*n-7)/2.
A088409: a(n) = A063416(n)/7.
A208688: T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 1 1 and 1 0 1 vertically
A241638: Number of partitions p of n such that (number of even numbers in p) = (number of odd numbers in p).
A325793: Positive integers whose number of divisors is equal to their sum of prime indices.
A351255: Numbers whose k-th arithmetic derivative is zero for some k>0, ordered by their position in A276086.

Number of the day: 545083

Properties of the number 545083:

545083 is a cyclic number.
545083 = 7 × 11 × 7079 is a sphenic number and squarefree.
545083 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 424680 totatives.
545083 has a semiprime digit sum 25 in base 10.
545083 = 272542² - 272541² = 38938² - 38931² = 24782² - 24771² = 3578² - 3501² is the difference of 2 nonnegative squares in 4 ways.
545083 is the sum of 2 positive triangular numbers.
545083 is the difference of 2 positive pentagonal numbers in 4 ways.
545083 = 85² + 123² + 723² is the sum of 3 positive squares.
545083² is the sum of 3 positive squares.
545083 is a proper divisor of 43⁷⁰⁷⁸ - 1.
545083 = '5' + '45083' is the concatenation of 2 prime numbers.

The number 545083 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A174728: Triangle read by rows: T(n, m, q) = (1-q^n)*Eulerian(n+1, m) - (1-q^n) + 1, with q = 2.

Number of the day: 165801

Charles Émile Picard was born on this day 166 years ago.

Properties of the number 165801:

165801 is a cyclic number.
165801 = 3 × 17 × 3251 is a sphenic number and squarefree.
165801 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 104000 totatives.
165801 has a semiprime digit sum 21 in base 10.
165801 has a Fibonacci digit sum 21 in base 10.
165801 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 165801 results in a semiprime.
165801 = 82901² - 82900² = 27635² - 27632² = 4885² - 4868² = 1651² - 1600² is the difference of 2 nonnegative squares in 4 ways.
165801 is the difference of 2 positive pentagonal numbers in 2 ways.
165801 = 5² + 76² + 400² is the sum of 3 positive squares.
165801² = 78024² + 146295² is the sum of 2 positive squares in 1 way.
165801² is the sum of 3 positive squares.
165801 is a proper divisor of 1237²⁰ - 1.
165801 is palindromic in (at least) the following bases: 50, -64, and -90.
165801 in base 50 = 1GG1 and consists of only the digits '1' and 'G'.

The number 165801 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A005904: Centered dodecahedral numbers.

Number of the day: 7222022

Happy Casual π Day!

Properties of the number 7222022:

7222022 = 2 × 43 × 79 × 1063 is the 6731307^th composite number and is squarefree.
7222022 has 4 distinct prime factors, 16 divisors, 27 antidivisors and 3479112 totatives.
7222022 has an emirp digit sum 17 in base 10.
7222022 = 17² + 42² + 2687² is the sum of 3 positive squares.
7222022² is the sum of 3 positive squares.
7222022 is a proper divisor of 7²³⁴ - 1.

Number of the day: 763856

Properties of the number 763856:

763856 = 2⁴ × 47741 is the 702587^th composite number and is not squarefree.
763856 has 2 distinct prime factors, 10 divisors, 9 antidivisors and 381920 totatives.
763856 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 763856 results in a prime.
763856 = 190965² - 190963² = 95484² - 95480² = 47745² - 47737² is the difference of 2 nonnegative squares in 3 ways.
763856 is the difference of 2 positive pentagonal numbers in 1 way.
763856 = 616² + 620² is the sum of 2 positive squares in 1 way.
763856 = 56² + 204² + 848² is the sum of 3 positive squares.
763856² = 4944² + 763840² is the sum of 2 positive squares in 1 way.
763856² is the sum of 3 positive squares.
763856 is a proper divisor of 1163¹⁴⁰ - 1.
763856 = '76385' + '6' is the concatenation of 2 semiprime numbers.

Number of the day: 258031

Properties of the number 258031:

258031 is a cyclic number.
258031 is the 22678^th prime.
258031 has 17 antidivisors and 258030 totatives.
258031 has a prime digit sum 19 in base 10.
258031 = 129016² - 129015² is the difference of 2 nonnegative squares in 1 way.
258031 is the difference of 2 positive pentagonal numbers in 1 way.
258031 is not the sum of 3 positive squares.
258031² is the sum of 3 positive squares.
258031 is a proper divisor of 919¹⁴¹ - 1.
258031 = '2' + '58031' is the concatenation of 2 prime numbers.
258031 = '25' + '8031' is the concatenation of 2 semiprime numbers.
258031 is an emirp in (at least) the following bases: 2, 9, 16, 19, 21, 22, 27, 28, 29, 31, 33, 45, 47, 52, 56, 60, 61, 63, 65, 66, 72, 74, 77, 79, 90, 91, 94, and 95.
258031 is palindromic in (at least) the following bases: 4, and -73.
258031 in base 4 = 332333233 and consists of only the digits '2' and '3'.

The number 258031 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A224605: Number of (n+2) X 9 0..2 matrices with each 3 X 3 subblock idempotent.
A254905: Number of (n+2)X(6+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, vertically and ne-to-sw antidiagonally
A254907: T(n,k)=Number of (n+2)X(k+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, vertically and ne-to-sw antidiagonally
A254908: Number of (2+2)X(n+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, vertically and ne-to-sw antidiagonally

Number of the day: 23823322

Properties of the number 23823322:

23823322 = 2 × 1123 × 10607 is a sphenic number and squarefree.
23823322 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 11899932 totatives.
23823322 has a semiprime digit sum 25 in base 10.
23823322 is the difference of 2 positive pentagonal numbers in 3 ways.
23823322 = 24² + 239² + 4875² is the sum of 3 positive squares.
23823322² is the sum of 3 positive squares.
23823322 is a proper divisor of 857⁵⁸³³³ - 1.
23823322 = '238233' + '22' is the concatenation of 2 semiprime numbers.
23823322 = '2382' + '3322' is the concatenation of 2 sphenic numbers.

Number of the day: 72737033

Properties of the number 72737033:

72737033 is a cyclic number.
72737033 = 17 × 4278649 is semiprime and squarefree.
72737033 has 2 distinct prime factors, 4 divisors, 37 antidivisors and 68458368 totatives.
Reversing the decimal digits of 72737033 results in an emirpimes.
72737033 = 36368517² - 36368516² = 2139333² - 2139316² is the difference of 2 nonnegative squares in 2 ways.
72737033 is the difference of 2 positive pentagonal numbers in 2 ways.
72737033 = 2248² + 8227² = 1888² + 8317² is the sum of 2 positive squares in 2 ways.
72737033 = 26² + 209² + 8526² is the sum of 3 positive squares.
72737033² = 36988592² + 62630025² = 34229192² + 64179735² = 31404992² + 65607945² = 3164040² + 72668183² is the sum of 2 positive squares in 4 ways.
72737033² is the sum of 3 positive squares.
72737033 is a proper divisor of 1931¹³⁶⁴⁸ - 1.
72737033 = '7273' + '7033' is the concatenation of 2 semiprime numbers.
72737033 is an emirpimes in (at least) the following bases: 6, 7, 10, 11, 14, 17, 20, 22, 27, 31, 33, 36, 37, 45, 47, 48, 49, 55, 56, 63, 68, 70, 75, 78, 82, 83, 84, 87, 89, 90, 95, 97, and 99.

Number of the day: 7172022

Happy Yellow Pig Day!

Properties of the number 7172022:

7172022 = 2 × 3 × 11 × 13² × 643 is the 6684450^th composite number and is not squarefree.
7172022 has 5 distinct prime factors, 48 divisors, 27 antidivisors and 2003040 totatives.
7172022 has a semiprime digit sum 21 in base 10.
7172022 has a Fibonacci digit sum 21 in base 10.
7172022 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 7172022 results in a sphenic number.
7172022 is the sum of 2 positive triangular numbers.
7172022 is the difference of 2 positive pentagonal numbers in 7 ways.
7172022 = 43² + 62² + 2677² is the sum of 3 positive squares.
7172022² = 2758470² + 6620328² = 5050122² + 5092560² is the sum of 2 positive squares in 2 ways.
7172022² is the sum of 3 positive squares.
7172022 is a proper divisor of 1109³⁹⁰ - 1.
7172022 in base 3 = 111111101011110 and consists of only the digits '0' and '1'.

Number of the day: 250162601912

Properties of the number 250162601912:

250162601912 = 2³ × 31 × 97 × 10399177 is composite and not squarefree.
250162601912 has 4 distinct prime factors, 32 divisors, 31 antidivisors and 119798507520 totatives.
250162601912 has a semiprime digit sum 35 in base 10.
250162601912 = 684³ + 2936³ + 6078³ = 3806³ + 4240³ + 4916³ is the sum of 3 positive cubes in 2 ways.
250162601912 = 62540650479² - 62540650477² = 31270325241² - 31270325237² = 2017440369² - 2017440307² = 1008720231² - 1008720107² = 644749071² - 644748877² = 322374681² - 322374293² = 20801361² - 20795347² = 10405191² - 10393163² is the difference of 2 nonnegative squares in 8 ways.
250162601912 = 466² + 3516² + 500150² is the sum of 3 positive squares.
250162601912² = 168618791040² + 184794563512² = 167634733240² + 185687704512² = 148905717688² + 201018443520² = 24175125312² + 248991748280² is the sum of 2 positive squares in 4 ways.
250162601912² is the sum of 3 positive squares.
250162601912 is a proper divisor of 1297⁸⁶⁶⁵⁹⁸ - 1.

Number of the day: 180293

Properties of the number 180293:

180293 is a cyclic number.
180293 = 29 × 6217 is semiprime and squarefree.
180293 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 174048 totatives.
180293 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 180293 results in an emirpimes.
180293 = 90147² - 90146² = 3123² - 3094² is the difference of 2 nonnegative squares in 2 ways.
180293 is the difference of 2 positive pentagonal numbers in 2 ways.
180293 is the difference of 2 positive pentagonal pyramidal numbers in 1 way.
180293 = 257² + 338² = 47² + 422² is the sum of 2 positive squares in 2 ways.
180293 = 40² + 63² + 418² is the sum of 3 positive squares.
180293² = 124340² + 130557² = 48195² + 173732² = 39668² + 175875² = 92568² + 154715² is the sum of 2 positive squares in 4 ways.
180293² is the sum of 3 positive squares.
180293 is a proper divisor of 1151¹⁴ - 1.
180293 is an emirpimes in (at least) the following bases: 4, 5, 7, 8, 9, 10, 13, 15, 21, 23, 24, 26, 29, 31, 34, 35, 36, 40, 42, 44, 45, 48, 49, 50, 51, 52, 55, 57, 59, 69, 75, 77, 82, 83, 92, 95, and 97.
180293 is palindromic in (at least) the following bases: -90, and -91.

The number 180293 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A146422: Number of n X n binary arrays symmetric under horizontal reflection with all ones connected only in a 1010-1111 pattern in any orientation.

Number of the day: 90251429

Properties of the number 90251429:

90251429 is a cyclic number.
90251429 = 89 × 1014061 is semiprime and squarefree.
90251429 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 89237280 totatives.
Reversing the decimal digits of 90251429 results in a prime.
90251429 = 45125715² - 45125714² = 507075² - 506986² is the difference of 2 nonnegative squares in 2 ways.
90251429 is the difference of 2 positive pentagonal numbers in 2 ways.
90251429 = 5390² + 7823² = 4670² + 8273² is the sum of 2 positive squares in 2 ways.
90251429 = 23² + 30² + 9500² is the sum of 3 positive squares.
90251429² = 46633629² + 77269820² = 39548379² + 81124880² = 32147229² + 84331940² = 8058060² + 89890979² is the sum of 2 positive squares in 4 ways.
90251429² is the sum of 3 positive squares.
90251429 is a proper divisor of 1301⁶⁷⁶⁰⁴ - 1.
90251429 is an emirpimes in (at least) the following bases: 11, 15, 17, 25, 26, 33, 34, 35, 38, 41, 42, 45, 47, 49, 57, 58, 70, 71, 73, 74, 75, 79, 82, 84, 85, 87, 88, 90, 94, 95, and 98.