Number of the day: 135327

Karl Weierstrass was born on this day 205 years ago.

Ronald Lewis Graham was born on this day 85 years ago.

Properties of the number 135327:

135327 = 3 × 79 × 571 is a sphenic number and squarefree.
135327 has 3 distinct prime factors, 8 divisors, 37 antidivisors and 88920 totatives.
135327 has a semiprime digit sum 21 in base 10.
135327 has a Fibonacci digit sum 21 in base 10.
135327 has a triangular digit sum 21 in base 10.
135327 has a triangular digit product 630 in base 10.
Reversing the decimal digits of 135327 results in a semiprime.
135327 = 67664² - 67663² = 22556² - 22553² = 896² - 817² = 404² - 167² is the difference of 2 nonnegative squares in 4 ways.
135327 is the sum of 2 positive triangular numbers.
135327 is the difference of 2 positive pentagonal numbers in 2 ways.
135327 is not the sum of 3 positive squares.
135327² is the sum of 3 positive squares.
135327 is a proper divisor of 103³⁰ - 1.
135327 in base 32 = 444v and consists of only the digits '4' and 'v'.
135327 in base 51 = 111O and consists of only the digits '1' and 'O'.
135327 in base 58 = eDD and consists of only the digits 'D' and 'e'.

The number 135327 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A331755: Number of vertices in a regular drawing of the complete bipartite graph K_{n,n}.

Number of the day: 171166

William Paul Thurston was born on this day 74 years ago.

Properties of the number 171166:

171166 = 2 × 23 × 61² is the 155567^th composite number and is not squarefree.
171166 has 3 distinct prime factors, 12 divisors, 21 antidivisors and 80520 totatives.
171166 has a semiprime digit sum 22 in base 10.
Reversing the decimal digits of 171166 results in a sphenic number.
171166 is the difference of 2 positive pentagonal numbers in 4 ways.
171166 = 9² + 146² + 387² is the sum of 3 positive squares.
171166² = 30866² + 168360² = 60720² + 160034² is the sum of 2 positive squares in 2 ways.
171166² is the sum of 3 positive squares.
171166 is a proper divisor of 601¹¹⁰ - 1.
171166 = '1711' + '66' is the concatenation of 2 triangular numbers.
171166 is palindromic in (at least) base 57.
171166 in base 57 = qcq and consists of only the digits 'c' and 'q'.
171166 in base 61 = k00 and consists of only the digits '0' and 'k'.

Number of the day: 880989807808

Properties of the number 880989807808:

880989807808 = 2⁶ × 11 × 41 × 30522097 is composite and not squarefree.
880989807808 has 4 distinct prime factors, 56 divisors, 39 antidivisors and 390682828800 totatives.
880989807808 has an emirp digit sum 73 in base 10.
880989807808 is the difference of 2 nonnegative squares in 20 ways.
880989807808 is the difference of 2 positive pentagonal numbers in 4 ways.
880989807808 = 6920² + 14568² + 938472² is the sum of 3 positive squares.
880989807808² = 193388006592² + 859502251520² = 185679752192² + 861200366400² = 201080737792² + 857735144640² = 7893005120² + 880954449408² is the sum of 2 positive squares in 4 ways.
880989807808² is the sum of 3 positive squares.
880989807808 is a proper divisor of 1663⁶⁴²³⁰ - 1.

Number of the day: 2664

Properties of the number 2664:

2664 is the 706^th totient number.
2664 = 2³ × 3² × 37 is the 2277^th composite number and is not squarefree.
2664 has 3 distinct prime factors, 24 divisors, 8 antidivisors and 864 totatives.
2664 = 667² - 665² = 335² - 331² = 225² - 219² = 117² - 105² = 83² - 65² = 55² - 19² is the difference of 2 nonnegative squares in 6 ways.
2664 is the sum of 2 positive triangular numbers.
2664 = 30² + 42² is the sum of 2 positive squares in 1 way.
2664 = 8² + 10² + 50² is the sum of 3 positive squares.
2664² = 864² + 2520² is the sum of 2 positive squares in 1 way.
2664² is the sum of 3 positive squares.
2664 is a proper divisor of 73² - 1.
2664 is palindromic in (at least) the following bases: 11, 71, 73, and -28.
2664 in base 6 = 20200 and consists of only the digits '0' and '2'.
2664 in base 11 = 2002 and consists of only the digits '0' and '2'.
2664 in base 19 = 774 and consists of only the digits '4' and '7'.
2664 in base 36 = 220 and consists of only the digits '0' and '2'.
2664 in base 51 = 11C and consists of only the digits '1' and 'C'.

The number 2664 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001631: Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4), with initial conditions a(0..3) = (0, 0, 1, 0).
A033586: a(n) = 4*n*(2*n + 1).
A046092: 4 times triangular numbers: a(n) = 2*n*(n+1).
A084921: a(n) = lcm(p-1, p+1) where p is the n-th prime.
A088557: Least even leg of primitive Pythagorean triangles with odd leg 2n+1.
A103854: Positive integers n such that n^6 + 1 is semiprime.
A219048: Numbers n such that 3^n + 32 is prime.
A268904: T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or antidiagonally adjacent neighbor totalling two exactly once.
A305148: Number of integer partitions of n whose distinct parts are pairwise indivisible.
A328171: Number of (necessarily strict) integer partitions of n with no two consecutive parts divisible.

Number of the day: 59651

Properties of the number 59651:

59651 is a cyclic number.
59651 is the 6029^th prime.
59651 has 33 antidivisors and 59650 totatives.
59651 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 59651 results in a sphenic number.
59651 = (138 × 139)/2 + … + (143 × 144)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
59651 = 29826² - 29825² is the difference of 2 nonnegative squares in 1 way.
59651 is the difference of 2 positive pentagonal numbers in 1 way.
59651 = 1² + 59² + 237² is the sum of 3 positive squares.
59651² is the sum of 3 positive squares.
59651 is a proper divisor of 17¹¹⁹³ - 1.
59651 is an emirp in (at least) the following bases: 7, 11, 19, 22, 23, 25, 31, 33, 36, 40, 50, 58, 59, 61, 65, 66, 72, 73, and 97.
59651 is palindromic in (at least) the following bases: 71, and -44.
59651 in base 40 = bBB and consists of only the digits 'B' and 'b'.

The number 59651 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A159071: Primes which are the sum of 6 consecutive triangular numbers A000217.

Number of the day: 590126

Ferdinand Georg Frobenius was born on this day 171 years ago.

Shiing-Shen Chern was born on this day 109 years ago.

Properties of the number 590126:

590126 = 2 × 109 × 2707 is a sphenic number and squarefree.
590126 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 292248 totatives.
590126 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 590126 results in a sphenic number.
590126 = 54² + 119² + 757² is the sum of 3 positive squares.
590126² = 324840² + 492674² is the sum of 2 positive squares in 1 way.
590126² is the sum of 3 positive squares.
590126 is a proper divisor of 1327³⁶ - 1.
590126 is palindromic in (at least) base -86.

Number of the day: 91200673718

Évariste Galois was born on this day 209 years ago.

Properties of the number 91200673718:

91200673718 = 2 × 7 × 11 × 349 × 1696883 is composite and squarefree.
91200673718 has 5 distinct prime factors, 32 divisors, 79 antidivisors and 35430896160 totatives.
91200673718 is the difference of 2 positive pentagonal numbers in 8 ways.
91200673718 = 1233² + 3202² + 301975² is the sum of 3 positive squares.
91200673718² = 47037596760² + 78134674618² is the sum of 2 positive squares in 1 way.
91200673718² is the sum of 3 positive squares.
91200673718 is a proper divisor of 1871^1077582 - 1.
91200673718 = '91200673' + '718' is the concatenation of 2 semiprime numbers.

Number of the day: 171274

Properties of the number 171274:

171274 = 2 × 29 × 2953 is a sphenic number and squarefree.
171274 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 82656 totatives.
171274 has a semiprime digit sum 22 in base 10.
Reversing the decimal digits of 171274 results in a semiprime.
171274 is the difference of 2 positive pentagonal numbers in 2 ways.
171274 = 243² + 335² = 75² + 407² is the sum of 2 positive squares in 2 ways.
171274 = 72² + 83² + 399² is the sum of 3 positive squares.
171274² = 118120² + 124026² = 61050² + 160024² = 53176² + 162810² = 73776² + 154570² is the sum of 2 positive squares in 4 ways.
171274² is the sum of 3 positive squares.
171274 is a proper divisor of 1277⁴¹ - 1.
171274 is palindromic in (at least) base -61.
171274 in base 13 = 5cc5c and consists of only the digits '5' and 'c'.
171274 in base 60 = lYY and consists of only the digits 'Y' and 'l'.

The number 171274 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A185324: E.g.f. log(1/(2-tan(x)-sec(x)).

Number of the day: 10232020

Happy Mole Day!

Properties of the number 10232020:

10232020 = 2² × 5 × 103 × 4967 is the 9553052^th composite number and is not squarefree.
10232020 has 4 distinct prime factors, 24 divisors, 35 antidivisors and 4052256 totatives.
10232020 has a semiprime digit sum 10 in base 10.
10232020 has a triangular digit sum 10 in base 10.
10232020 = 2558006² - 2558004² = 511606² - 511596² = 24938² - 24732² = 5482² - 4452² is the difference of 2 nonnegative squares in 4 ways.
10232020 is the sum of 2 positive triangular numbers.
10232020 is the difference of 2 positive pentagonal numbers in 3 ways.
10232020 = 84² + 190² + 3192² is the sum of 3 positive squares.
10232020² = 6139212² + 8185616² is the sum of 2 positive squares in 1 way.
10232020² is the sum of 3 positive squares.
10232020 is a proper divisor of 307²⁶⁵² - 1.

Number of the day: 2243993847

Properties of the number 2243993847:

2243993847 = 3 × 457 × 1636757 is a sphenic number and squarefree.
2243993847 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 1492721472 totatives.
2243993847 has an emirpimes digit sum 51 in base 10.
Reversing the decimal digits of 2243993847 results in a sphenic number.
2243993847 = 1121996924² - 1121996923² = 373998976² - 373998973² = 2455364² - 2454907² = 819064² - 817693² is the difference of 2 nonnegative squares in 4 ways.
2243993847 is the sum of 2 positive triangular numbers.
2243993847 is the difference of 2 positive pentagonal numbers in 2 ways.
2243993847 is not the sum of 3 positive squares.
2243993847² = 1276632300² + 1845458847² = 824925528² + 2086865175² = 330245700² + 2219559903² = 508821972² + 2185545375² is the sum of 2 positive squares in 4 ways.
2243993847² is the sum of 3 positive squares.
2243993847 is a proper divisor of 109¹⁴⁸⁷⁹⁶ - 1.
2243993847 = '22439' + '93847' is the concatenation of 2 semiprime numbers.

Number of the day: 568856135706

Martin Gardner was born on this day 106 years ago.

Properties of the number 568856135706:

568856135706 = 2 × 3 × 31 × 83 × 563 × 65449 is composite and squarefree.
568856135706 has 6 distinct prime factors, 64 divisors, 39 antidivisors and 180966337920 totatives.
568856135706 is the difference of 2 positive pentagonal numbers in 8 ways.
568856135706 = 1144² + 1579² + 754223² is the sum of 3 positive squares.
568856135706² = 337929174720² + 457603732506² is the sum of 2 positive squares in 1 way.
568856135706² is the sum of 3 positive squares.
568856135706 is a proper divisor of 1327^1532574 - 1.
568856135706 = '56885' + '6135706' is the concatenation of 2 sphenic numbers.

Number of the day: 5020

Properties of the number 5020:

5020 is the 1268^th totient number.
5020 and 5564 are a pair of amicable numbers.
5020 = 2² × 5 × 251 is the 4347^th composite number and is not squarefree.
5020 has 3 distinct prime factors, 12 divisors, 5 antidivisors and 2000 totatives.
5020 has a prime digit sum 7 in base 10.
5020 = 1256² - 1254² = 256² - 246² is the difference of 2 nonnegative squares in 2 ways.
5020 is the difference of 2 positive pentagonal numbers in 2 ways.
5020 is the difference of 2 positive pentagonal pyramidal numbers in 1 way.
5020 is not the sum of 3 positive squares.
5020² = 3012² + 4016² is the sum of 2 positive squares in 1 way.
5020² is the sum of 3 positive squares.
5020 is a proper divisor of 503⁴ - 1.
5020 is palindromic in (at least) the following bases: 33, -18, and -38.
5020 in base 32 = 4ss and consists of only the digits '4' and 's'.
5020 in base 33 = 4k4 and consists of only the digits '4' and 'k'.

The number 5020 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001466: Denominators of greedy Egyptian fraction expansion of Pi - 3.
A002025: Smaller of an amicable pair: (a,b) such that sigma(a)=sigma(b)=a+b, a<b.
A063990: Amicable numbers.
A158652: Any two consecutive digits in the sequence sum up to a prime.
A188236: T(n,k)=Number of nondecreasing arrangements of n numbers in -(n+k-2)..(n+k-2) with sum zero and not more than two numbers equal
A256631: Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 5 as largest digit.
A259180: Amicable pairs.
A259933: Amicable pairs (x < y) ordered by nondecreasing sum (x + y) and then by increasing x.
A276091: Numbers obtained by reinterpreting base-2 representation of n in A001563-base (A276326): a(n) = Sum_k>=0 {A030308(n,k)*A001563(k+1)}.
A316724: Generalized 26-gonal (or icosihexagonal) numbers: m*(12*m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...

Number of the day: 271760

Properties of the number 271760:

271760 = 2⁴ × 5 × 43 × 79 is the 247975^th composite number and is not squarefree.
271760 has 4 distinct prime factors, 40 divisors, 19 antidivisors and 104832 totatives.
271760 has a prime digit sum 23 in base 10.
271760 = (114 × 115)/2 + … + (145 × 146)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
271760 is the difference of 2 nonnegative squares in 12 ways.
271760 is the difference of 2 positive pentagonal numbers in 4 ways.
271760 = 20² + 96² + 512² is the sum of 3 positive squares.
271760² = 163056² + 217408² is the sum of 2 positive squares in 1 way.
271760² is the sum of 3 positive squares.
271760 is a proper divisor of 947⁴ - 1.

The number 271760 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A200379: Second row of array defined in A056230.

Number of the day: 603210

Properties of the number 603210:

603210 = 2 × 3 × 5 × 20107 is the 553864^th composite number and is squarefree.
603210 has 4 distinct prime factors, 16 divisors, 11 antidivisors and 160848 totatives.
603210 has an oblong digit sum 12 in base 10.
603210 is the difference of 2 positive pentagonal numbers in 2 ways.
603210 = 40² + 169² + 757² is the sum of 3 positive squares.
603210² = 361926² + 482568² is the sum of 2 positive squares in 1 way.
603210² is the sum of 3 positive squares.
603210 is a proper divisor of 811¹¹¹⁷ - 1.
603210 is palindromic in (at least) base 97.

Number of the day: 514385

Properties of the number 514385:

514385 is a cyclic number.
514385 = 5 × 102877 is semiprime and squarefree.
514385 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 411504 totatives.
514385 has an emirpimes digit sum 26 in base 10.
514385 = 257193² - 257192² = 51441² - 51436² is the difference of 2 nonnegative squares in 2 ways.
514385 is the difference of 2 positive pentagonal numbers in 2 ways.
514385 = 481² + 532² = 137² + 704² is the sum of 2 positive squares in 2 ways.
514385 = 64² + 95² + 708² is the sum of 3 positive squares.
514385² = 308631² + 411508² = 192896² + 476847² = 51663² + 511784² = 265740² + 440425² is the sum of 2 positive squares in 4 ways.
514385² is the sum of 3 positive squares.
514385 is a proper divisor of 11⁸⁵⁷³ - 1.
514385 = '51' + '4385' is the concatenation of 2 emirpimes.
514385 is an emirpimes in (at least) the following bases: 3, 4, 5, 6, 7, 8, 12, 14, 15, 16, 21, 22, 25, 28, 30, 33, 34, 35, 38, 46, 50, 52, 55, 57, 60, 65, 68, 71, 75, 76, 79, 85, 86, 89, 91, 93, 95, and 96.

Number of the day: 7545202826

Properties of the number 7545202826:

7545202826 = 2 × 7 × 538943059 is a sphenic number and squarefree.
7545202826 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 3233658348 totatives.
7545202826 has a prime digit sum 41 in base 10.
Reversing the decimal digits of 7545202826 results in a sphenic number.
7545202826 = 269² + 996² + 86857² is the sum of 3 positive squares.
7545202826² is the sum of 3 positive squares.
7545202826 is a proper divisor of 809^1852038 - 1.
7545202826 = '7545202' + '826' is the concatenation of 2 sphenic numbers.

Number of the day: 93556

Properties of the number 93556:

93556 = 2² × 19 × 1231 is the 84521^th composite number and is not squarefree.
93556 has 3 distinct prime factors, 12 divisors, 9 antidivisors and 44280 totatives.
93556 has a triangular digit sum 28 in base 10.
Reversing the decimal digits of 93556 results in a prime.
93556 = 23390² - 23388² = 1250² - 1212² is the difference of 2 nonnegative squares in 2 ways.
93556 is the sum of 2 positive triangular numbers.
93556 is the difference of 2 positive pentagonal numbers in 2 ways.
93556 = 44² + 192² + 234² is the sum of 3 positive squares.
93556² is the sum of 3 positive squares.
93556 is a proper divisor of 761¹⁵ - 1.
93556 = '9355' + '6' is the concatenation of 2 semiprime numbers.
93556 is palindromic in (at least) the following bases: 15, and 88.
93556 in base 45 = 1191 and consists of only the digits '1' and '9'.

The number 93556 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A299475: a(n) is the number of vertices in the diagram of partitions of n (see example).
A308989: Sum of all the parts in the partitions of n into 8 parts.

Number of the day: 3668109

Properties of the number 3668109:

3668109 = 3 × 23 × 53161 is a sphenic number and squarefree.
3668109 has 3 distinct prime factors, 8 divisors, 23 antidivisors and 2339040 totatives.
3668109 has a semiprime digit sum 33 in base 10.
Reversing the decimal digits of 3668109 results in a sphenic number.
3668109 = 1834055² - 1834054² = 611353² - 611350² = 79753² - 79730² = 26615² - 26546² is the difference of 2 nonnegative squares in 4 ways.
3668109 is the difference of 2 positive pentagonal numbers in 2 ways.
3668109 = 130² + 203² + 1900² is the sum of 3 positive squares.
3668109² = 2094840² + 3011091² is the sum of 2 positive squares in 1 way.
3668109² is the sum of 3 positive squares.
3668109 is a proper divisor of 191⁴⁴⁰ - 1.
3668109 is palindromic in (at least) base 31.
3668109 in base 31 = 3u3u3 and consists of only the digits '3' and 'u'.

Number of the day: 441327

Properties of the number 441327:

441327 = 3 × 157 × 937 is a sphenic number and squarefree.
441327 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 292032 totatives.
441327 has a semiprime digit sum 21 in base 10.
441327 has a Fibonacci digit sum 21 in base 10.
441327 has a triangular digit sum 21 in base 10.
441327 = 220664² - 220663² = 73556² - 73553² = 1484² - 1327² = 704² - 233² is the difference of 2 nonnegative squares in 4 ways.
441327 is the difference of 2 positive pentagonal numbers in 2 ways.
441327 is not the sum of 3 positive squares.
441327² = 238935² + 371052² = 147420² + 415977² = 306327² + 317700² = 101265² + 429552² is the sum of 2 positive squares in 4 ways.
441327² is the sum of 3 positive squares.
441327 is a proper divisor of 13¹⁸ - 1.
441327 is palindromic in (at least) base -25.

Number of the day: 9913

Properties of the number 9913:

9913 is a cyclic number.
9913 = 23 × 431 is semiprime and squarefree.
9913 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 9460 totatives.
9913 has a semiprime digit sum 22 in base 10.
Reversing the decimal digits of 9913 results in an emirpimes.
9913 = 4957² - 4956² = 227² - 204² is the difference of 2 nonnegative squares in 2 ways.
9913 is the difference of 2 positive pentagonal numbers in 2 ways.
9913 = 16² + 51² + 84² is the sum of 3 positive squares.
9913² is the sum of 3 positive squares.
9913 is a proper divisor of 863¹¹ - 1.
9913 = '991' + '3' is the concatenation of 2 prime numbers.
9913 = '9' + '913' is the concatenation of 2 semiprime numbers.
9913 is an emirpimes in (at least) the following bases: 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 16, 19, 24, 25, 29, 32, 37, 41, 44, 45, 52, 59, 65, 66, 73, 74, 77, 80, 81, 85, 87, 91, 92, 96, 97, 98, and 99.
9913 is palindromic in (at least) the following bases: 35, 84, and -39.
9913 in base 21 = 11a1 and consists of only the digits '1' and 'a'.
9913 in base 34 = 8jj and consists of only the digits '8' and 'j'.
9913 in base 35 = 838 and consists of only the digits '3' and '8'.
9913 in base 44 = 55D and consists of only the digits '5' and 'D'.

The number 9913 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A007811: Numbers n for which 10n+1, 10n+3, 10n+7 and 10n+9 are primes.
A080697: a(n) = n * prime(prime(n)).
A118367: Largest number that is not the sum of five n-gonal numbers.
A208807: Number of n-bead necklaces labeled with numbers -1..1 not allowing reversal, with sum zero and three times sum of squares <= n*(1)*(1+1).
A233585: Coefficients of the generalized continued fraction expansion of the inverse of Euler constant, 1/gamma = a(1) +a(1)/(a(2) +a(2)/(a(3) +a(3)/(a(4) +a(4)/....))).
A244791: Number of length 3 0..n arrays with each partial sum starting from the beginning no more than one standard deviation from its mean.
A256533: Product of n and the sum of all divisors of all positive integers <= n.
A270893: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 209", based on the 5-celled von Neumann neighborhood.
A284648: Numerator of sum of reciprocals of all divisors of all positive integers <= n.
A331453: Triangle read by rows: T(n,m) (n >= m >= 1) = number of vertices formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

Number of the day: 603423725152

Properties of the number 603423725152:

603423725152 = 2⁵ × 67 × 281 × 1001593 is composite and not squarefree.
603423725152 has 4 distinct prime factors, 48 divisors, 83 antidivisors and 296150722560 totatives.
603423725152 is the difference of 2 nonnegative squares in 16 ways.
603423725152 is the difference of 2 positive pentagonal numbers in 4 ways.
603423725152 = 3976² + 10380² + 776724² is the sum of 3 positive squares.
603423725152² = 343586462720² + 496052955552² = 321532045920² + 510624456448² = 364981556640² + 480529557248² = 26427083360² + 602844757248² is the sum of 2 positive squares in 4 ways.
603423725152² is the sum of 3 positive squares.
603423725152 is a proper divisor of 1409⁸³¹⁶⁰ - 1.

Number of the day: 90131

Properties of the number 90131:

90131 is a cyclic number.
90131 = 193 × 467 is semiprime and squarefree.
90131 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 89472 totatives.
90131 has a semiprime digit sum 14 in base 10.
Reversing the decimal digits of 90131 results in a prime.
90131 = 45066² - 45065² = 330² - 137² is the difference of 2 nonnegative squares in 2 ways.
90131 is the difference of 2 positive pentagonal numbers in 1 way.
90131 = 9² + 55² + 295² is the sum of 3 positive squares.
90131² = 44365² + 78456² is the sum of 2 positive squares in 1 way.
90131² is the sum of 3 positive squares.
90131 is a proper divisor of 1867¹² - 1.
90131 is an emirpimes in (at least) the following bases: 2, 3, 4, 9, 13, 16, 23, 25, 29, 31, 32, 36, 39, 42, 43, 44, 51, 55, 59, 60, 62, 64, 65, 66, 67, 75, 77, 87, 91, 93, 97, and 98.
90131 is palindromic in (at least) base 82.

The number 90131 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A207059: Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+289)^2 = y^2.
A307881: 2*a(n) is the least number where k sets a new record such that 2*a(n)-k and 2*a(n)+k are prime and at least one of 2*a(n)-j and 2*a(n)+j is composite for all 0<j<k.
A328466: Row 3 of A328464: a(n) = A276156(8n - 4) / 6.

Number of the day: 9781

Properties of the number 9781:

9781 is a cyclic number.
9781 is the 1206^th prime.
9781 has 5 antidivisors and 9780 totatives.
9781 has a semiprime digit sum 25 in base 10.
9781 has sum of divisors equal to 9782 which is a sphenic number.
Reversing the decimal digits of 9781 results in an emirp.
9781 = 2³ + 8³ + 21³ is the sum of 3 positive cubes in 1 way.
9781 = 4891² - 4890² is the difference of 2 nonnegative squares in 1 way.
9781 is the sum of 2 positive triangular numbers.
9781 is the difference of 2 positive pentagonal numbers in 1 way.
9781 = 41² + 90² is the sum of 2 positive squares in 1 way.
9781 = 36² + 49² + 78² is the sum of 3 positive squares.
9781² = 6419² + 7380² is the sum of 2 positive squares in 1 way.
9781² is the sum of 3 positive squares.
9781 is a proper divisor of 359¹⁰ - 1.
9781 = '9' + '781' is the concatenation of 2 semiprime numbers.
9781 is an emirp in (at least) the following bases: 6, 9, 10, 15, 17, 18, 19, 20, 21, 27, 31, 36, 37, 43, 44, 52, 55, 56, 59, 69, 70, 71, 75, 76, 78, 83, 84, 86, 87, 89, 93, 94, 95, and 99.
9781 is palindromic in (at least) the following bases: -47, and -77.

The number 9781 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A074341: a(1) = 4; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A085013: a(1) = 1; for n>1, a(n) = smallest prime > a(n-1) such that a(1)*...*a(n) + 2 is a prime.
A089442: Primes p such that (p-11)/10 is also a prime.
A139489: Primes of the form x^2+101y^2.
A142122: Primes congruent to 13 mod 37.
A199936: Total sum of Fibonacci parts in all partitions of n.
A206610: Fibonacci sequence beginning 13, 8.
A262508: Numbers that occur only once in A155043; positions of zeros in A262505, ones in A262507.
A299110: Prime(r) for r such that prime(r) - prime(r-1) = 12 and prime(r-1) - prime(r-2) = 2.
A300334: Primes of a056240-type 2.

Number of the day: 4903

Properties of the number 4903:

4903 is a cyclic number.
4903 is the 655^th prime.
4903 has 13 antidivisors and 4902 totatives.
4903 = 2452² - 2451² is the difference of 2 nonnegative squares in 1 way.
4903 is the difference of 2 positive pentagonal numbers in 1 way.
4903 is not the sum of 3 positive squares.
4903² is the sum of 3 positive squares.
4903 is a proper divisor of 373³⁸ - 1.
4903 is an emirp in (at least) the following bases: 2, 3, 4, 14, 15, 17, 21, 26, 29, 38, 41, 45, 50, 56, 59, 60, 61, 69, 71, 79, 80, 83, 85, 87, 91, 99, and 100.
4903 is palindromic in (at least) the following bases: 31, 57, -49, and -86.
4903 in base 17 = gg7 and consists of only the digits '7' and 'g'.
4903 in base 30 = 5dd and consists of only the digits '5' and 'd'.
4903 in base 31 = 535 and consists of only the digits '3' and '5'.
4903 in base 49 = 223 and consists of only the digits '2' and '3'.
4903 in base 56 = 1VV and consists of only the digits '1' and 'V'.
4903 in base 57 = 1T1 and consists of only the digits '1' and 'T'.

The number 4903 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005347: First differences of A005579.
A033548: Honaker primes: primes P(k) such that sum of digits of P(k) equals sum of digits of k.
A049423: Primes of the form k^2 + 3.
A074339: a(1) = 3; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A112847: Primes such that the sum of the predecessor and successor primes is divisible by 23.
A139253: Toothpick primes: primes in the toothpick sequence A139250.
A141978: Primes congruent to 2 mod 29.
A153213: Primes p such that both p-2 and p+2 are not squarefree.
A154834: Primes p such that p^5-2 is also prime.
A183517: T(n,k)=Number of nXk binary arrays with each element equal to either the sum mod 2 of its horizontal and vertical neighbors or the sum mod 2 of its diagonal and antidiagonal neighbors

Number of the day: 71553

Richard Dedekind was born on this day 189 years ago.

Properties of the number 71553:

71553 = 3 × 17 × 23 × 61 is the 64465^th composite number and is squarefree.
71553 has 4 distinct prime factors, 16 divisors, 17 antidivisors and 42240 totatives.
71553 has a semiprime digit sum 21 in base 10.
71553 has a Fibonacci digit sum 21 in base 10.
71553 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 71553 results in a semiprime.
71553 = 35777² - 35776² = 11927² - 11924² = 2113² - 2096² = 1567² - 1544² = 727² - 676² = 617² - 556² = 553² - 484² = 287² - 104² is the difference of 2 nonnegative squares in 8 ways.
71553 is the sum of 2 positive triangular numbers.
71553 is the difference of 2 positive pentagonal numbers in 4 ways.
71553 = 11² + 26² + 266² is the sum of 3 positive squares.
71553² = 33672² + 63135² = 21735² + 68172² = 44505² + 56028² = 12903² + 70380² is the sum of 2 positive squares in 4 ways.
71553² is the sum of 3 positive squares.
71553 is a proper divisor of 599⁴ - 1.
71553 = '7' + '1553' is the concatenation of 2 prime numbers.
71553 is palindromic in (at least) the following bases: 73, and 88.
71553 in base 6 = 1311133 and consists of only the digits '1' and '3'.
71553 in base 53 = PP3 and consists of only the digits '3' and 'P'.

The number 71553 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A305262: a(n) = 140*2^n - 127.

Number of the day: 49923

Properties of the number 49923:

49923 = 3³ × 43² is the 44797^th composite number and is not squarefree.
49923 has 2 distinct prime factors, 12 divisors, 23 antidivisors and 32508 totatives.
49923 = 24962² - 24961² = 8322² - 8319² = 2778² - 2769² = 938² - 911² = 602² - 559² = 258² - 129² is the difference of 2 nonnegative squares in 6 ways.
49923 is the sum of 2 positive triangular numbers.
49923 is the difference of 2 positive pentagonal numbers in 1 way.
49923 = 5² + 13² + 223² is the sum of 3 positive squares.
49923² is the sum of 3 positive squares.
49923 is a proper divisor of 19⁴² - 1.
49923 = '499' + '23' is the concatenation of 2 prime numbers.
49923 = '49' + '923' is the concatenation of 2 emirpimes.
49923 is palindromic in (at least) the following bases: -62, -68, and -71.
49923 in base 4 = 30030003 and consists of only the digits '0' and '3'.
49923 in base 39 = WW3 and consists of only the digits '3' and 'W'.
49923 in base 43 = R00 and consists of only the digits '0' and 'R'.
49923 in base 61 = DPP and consists of only the digits 'D' and 'P'.

The number 49923 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A036314: Composite numbers n such that digits of prime factors of n are either 3 or 4.
A100450: Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
A135488: Number of distinct self-dual normal bases for GF(2^n) over GF(2)
A160228: Write Pi-3 in binary and report the number of zeros in the first 10^n decimal places.
A163787: a(n) is the n-th J_7-prime (Josephus_7 prime).
A242381: Positive integers, c, such that there are more than two solutions to the equation a^2 + b^3 = c^4, with a, b > 0.
A242867: Discriminants of cubic domains for cubefree n.

Number of the day: 773

Properties of the number 773:

773 is a cyclic number.
773 is the 137^th prime.
773 has 13 antidivisors and 772 totatives.
773 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 773 results in a semiprime.
Reversing the decimal digits of 773 results in a Fibonacci number.
773 = 387² - 386² is the difference of 2 nonnegative squares in 1 way.
773 is the difference of 2 positive pentagonal numbers in 1 way.
773 = 17² + 22² is the sum of 2 positive squares in 1 way.
773 = 4² + 9² + 26² is the sum of 3 positive squares.
773² = 195² + 748² is the sum of 2 positive squares in 1 way.
773² is the sum of 3 positive squares.
773 is a proper divisor of 317⁴ - 1.
773 = '7' + '73' is the concatenation of 2 prime numbers.
773 is an emirp in (at least) the following bases: 2, 3, 5, 7, 8, 13, 16, 19, 26, 27, 28, 29, 35, 41, 43, 44, 49, 53, 55, 59, 67, 69, 71, 77, 82, 89, 99, and 100.
773 is palindromic in (at least) the following bases: 12, and 14.
773 consists of only the digits '3' and '7'.
773 in base 12 = 545 and consists of only the digits '4' and '5'.
773 in base 14 = 3d3 and consists of only the digits '3' and 'd'.
773 in base 19 = 22d and consists of only the digits '2' and 'd'.
773 in base 27 = 11h and consists of only the digits '1' and 'h'.

The number 773 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000078: Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) for n >= 4 with a(0) = a(1) = a(2) = 0 and a(3) = 1.
A000928: Irregular primes: primes p such that at least one of the numerators of the Bernoulli numbers B_2, B_4, ..., B_{p-3} (A000367) is divisible by p.
A001122: Primes with primitive root 2.
A005235: Fortunate numbers: least m > 1 such that m + prime(n)# is prime, where p# denotes the product of all primes <= p.
A006450: Prime-indexed primes: primes with prime subscripts.
A007635: Primes of form n^2 + n + 17.
A019546: Primes whose digits are primes.
A037274: Home primes: for n >= 2, a(n) = the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached (a(n) = -1 if no prime is ever reached).
A046132: Larger member p+4 of cousin primes (p, p+4).
A299259: Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.8.8 2D tiling (cf. A008576).

Number of the day: 8323

Properties of the number 8323:

8323 = 7 × 29 × 41 is a sphenic number and squarefree.
8323 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 6720 totatives.
8323 has a Fibonacci digit product 144 in base 10.
Reversing the decimal digits of 8323 results in a semiprime.
8323 = (17 × 18)/2 + … + (37 × 38)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
8323 = 4162² - 4161² = 598² - 591² = 158² - 129² = 122² - 81² is the difference of 2 nonnegative squares in 4 ways.
8323 is the sum of 2 positive triangular numbers.
8323 is the difference of 2 positive pentagonal numbers in 3 ways.
8323 = 9² + 41² + 81² is the sum of 3 positive squares.
8323² = 5740² + 6027² = 4277² + 7140² = 4620² + 6923² = 1827² + 8120² is the sum of 2 positive squares in 4 ways.
8323² is the sum of 3 positive squares.
8323 is a proper divisor of 811⁴ - 1.
8323 = '83' + '23' is the concatenation of 2 prime numbers.
8323 is palindromic in (at least) the following bases: 33, 52, 53, 73, -19, -36, -47, and -64.
8323 in base 13 = 3a33 and consists of only the digits '3' and 'a'.
8323 in base 19 = 1411 and consists of only the digits '1' and '4'.
8323 in base 21 = ii7 and consists of only the digits '7' and 'i'.
8323 in base 27 = bb7 and consists of only the digits '7' and 'b'.
8323 in base 33 = 7l7 and consists of only the digits '7' and 'l'.
8323 in base 45 = 44h and consists of only the digits '4' and 'h'.
8323 in base 51 = 3AA and consists of only the digits '3' and 'A'.
8323 in base 52 = 343 and consists of only the digits '3' and '4'.
8323 in base 53 = 2p2 and consists of only the digits '2' and 'p'.

The number 8323 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A018018: Powers of cube root of 15 rounded down.
A018019: Powers of cube root of 15 rounded to nearest integer.
A071398: Rounded total surface area of a regular icosahedron with edge length n.
A075110: Concatenation of n-th prime and n in decimal notation.
A121733: Numbers k such that tau(k) = tau(k+1) mod 691, where tau is Ramanujan's tau function A000594.
A147874: a(n) = (5*n-7)*(n-1).
A171155: For two strings of length n, this is the number of pairwise alignments that do not have an insertion adjacent to a deletion.
A265064: Coordination sequence for (2,5,5) tiling of hyperbolic plane.
A276600: Values of n such that n^2 + 6 is a triangular number (A000217).
A324315: Squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p.

Number of the day: 433608

Properties of the number 433608:

433608 = 2³ × 3 × 7 × 29 × 89 is the 397156^th composite number and is not squarefree.
433608 has 5 distinct prime factors, 64 divisors, 27 antidivisors and 118272 totatives.
433608 is the difference of 2 nonnegative squares in 16 ways.
433608 is the difference of 2 positive pentagonal numbers in 2 ways.
433608 = 58² + 88² + 650² is the sum of 3 positive squares.
433608² = 299040² + 313992² = 131208² + 413280² = 151200² + 406392² = 190008² + 389760² is the sum of 2 positive squares in 4 ways.
433608² is the sum of 3 positive squares.
433608 is a proper divisor of 853⁸ - 1.

Number of the day: 84967

Properties of the number 84967:

84967 is a cyclic number.
84967 is the 8274^th prime.
84967 has 7 antidivisors and 84966 totatives.
84967 has a semiprime digit sum 34 in base 10.
84967 has a Fibonacci digit sum 34 in base 10.
84967 = 42484² - 42483² is the difference of 2 nonnegative squares in 1 way.
84967 is the sum of 2 positive triangular numbers.
84967 is the difference of 2 positive pentagonal numbers in 1 way.
84967 is not the sum of 3 positive squares.
84967² is the sum of 3 positive squares.
84967 is a proper divisor of 773⁴² - 1.
84967 is an emirp in (at least) the following bases: 4, 5, 15, 27, 29, 33, 36, 39, 40, 42, 50, 67, 70, 80, 87, 95, and 99.
84967 is palindromic in (at least) the following bases: 17, 47, 74, and -17.
84967 in base 47 = cLc and consists of only the digits 'L' and 'c'.
84967 in base 59 = OO7 and consists of only the digits '7' and 'O'.

The number 84967 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A023316: Numbers n such that n remains prime through 4 iterations of function f(x) = 5x + 8.
A082113: n^4+5n^2+1.
A090687: Primes of the form 6*k^2 + 1.
A283791: List of prime numbers p such that all prime factors of p+1 and p-1 are smaller than the cube root of p.
A322405: Number of compositions of n into parts 1, 8, 9.