Number of the day: 815484

Properties of the number 815484:

815484 = 2² × 3 × 67957 is the 750399^th composite number and is not squarefree.
815484 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 271824 totatives.
815484 has a sphenic digit sum 30 in base 10.
815484 has an oblong digit sum 30 in base 10.
815484 = 203872² - 203870² = 67960² - 67954² is the difference of 2 nonnegative squares in 2 ways.
815484 is the difference of 2 positive pentagonal numbers in 1 way.
815484 is not the sum of 3 positive squares.
815484² = 154140² + 800784² is the sum of 2 positive squares in 1 way.
815484² is the sum of 3 positive squares.
815484 is a proper divisor of 769⁴² - 1.
815484 is palindromic in (at least) the following bases: 16, and -73.

Number of the day: 10845

Properties of the number 10845:

10845 = 3² × 5 × 241 is the 9527^th composite number and is not squarefree.
10845 has 3 distinct prime factors, 12 divisors, 17 antidivisors and 5760 totatives.
10845 = 5423² - 5422² = 1809² - 1806² = 1087² - 1082² = 607² - 598² = 369² - 354² = 143² - 98² is the difference of 2 nonnegative squares in 6 ways.
10845 is the difference of 2 positive pentagonal numbers in 2 ways.
10845 = 69² + 78² = 21² + 102² is the sum of 2 positive squares in 2 ways.
10845 = 2² + 5² + 104² is the sum of 3 positive squares.
10845² = 6507² + 8676² = 4284² + 9963² = 1323² + 10764² = 5400² + 9405² is the sum of 2 positive squares in 4 ways.
10845² is the sum of 3 positive squares.
10845 is a proper divisor of 739⁶ - 1.
10845 is palindromic in (at least) base -36.
10845 in base 15 = 3330 and consists of only the digits '0' and '3'.
10845 in base 42 = 669 and consists of only the digits '6' and '9'.
10845 in base 46 = 55Z and consists of only the digits '5' and 'Z'.

The number 10845 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A022466: Number of 1's in n-th term of A007651.
A023627: Convolution of (1, p(1), p(2), ...) and composite numbers.
A069927: Numbers n such that n divides 2^(n+3)-1.
A143448: Expansion of 1/(x^k*(1-x-2*x^(k+1))) for k=5.
A151745: Composites that are the sum of two, three, four and five consecutive composite numbers.
A156370: Numerator of Euler(n, 5/14).
A201365: Expansion of e.g.f.: exp(x) / (5 - 4*exp(x)).
A236370: Sum of the largest parts in the partitions of 3n into 3 parts.
A295993: Numbers k such that there are precisely 8 groups of orders k and k + 1.
A321960: Array of sequences read by descending antidiagonals, A(n) the Jacobi square of the sequence n, n+1, n+2, ....

Number of the day: 58220

Ernst Eduard Kummer was born on this day 211 years ago.

Properties of the number 58220:

58220 = 2² × 5 × 41 × 71 is the 52322^th composite number and is not squarefree.
58220 has 4 distinct prime factors, 24 divisors, 19 antidivisors and 22400 totatives.
58220 has an emirp digit sum 17 in base 10.
58220 = 14556² - 14554² = 2916² - 2906² = 396² - 314² = 276² - 134² is the difference of 2 nonnegative squares in 4 ways.
58220 is the difference of 2 positive pentagonal numbers in 4 ways.
58220 = 26² + 30² + 238² is the sum of 3 positive squares.
58220² = 34932² + 46576² = 23856² + 53108² = 37772² + 44304² = 12780² + 56800² is the sum of 2 positive squares in 4 ways.
58220² is the sum of 3 positive squares.
58220 is a proper divisor of 409¹⁰ - 1.
58220 is palindromic in (at least) the following bases: 68, and 84.
58220 in base 58 = HHk and consists of only the digits 'H' and 'k'.

The number 58220 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A232967: Irregular triangle read by rows: row n lists the rank sizes of the "electrical" poset EP_n of circular planar graphs with n boundary vertices.

Number of the day: 82797

Louis J. Mordell was born on this day 133 years ago.

Properties of the number 82797:

82797 = 3 × 11 × 13 × 193 is the 74704^th composite number and is squarefree.
82797 has 4 distinct prime factors, 16 divisors, 19 antidivisors and 46080 totatives.
82797 has a semiprime digit sum 33 in base 10.
82797 = 41399² - 41398² = 13801² - 13798² = 3769² - 3758² = 3191² - 3178² = 1271² - 1238² = 1081² - 1042² = 361² - 218² = 311² - 118² is the difference of 2 nonnegative squares in 8 ways.
82797 is the difference of 2 positive pentagonal numbers in 3 ways.
82797 = 5² + 46² + 284² is the sum of 3 positive squares.
82797² = 31845² + 76428² = 9900² + 82203² = 50853² + 65340² = 40755² + 72072² is the sum of 2 positive squares in 4 ways.
82797² is the sum of 3 positive squares.
82797 is a proper divisor of 857⁶ - 1.
82797 = '827' + '97' is the concatenation of 2 prime numbers.
82797 is palindromic in (at least) base 73.
82797 in base 44 = gXX and consists of only the digits 'X' and 'g'.

Number of the day: 995338558

Properties of the number 995338558:

995338558 = 2 × 911 × 546289 is a sphenic number and squarefree.
995338558 has 3 distinct prime factors, 8 divisors, 55 antidivisors and 497122080 totatives.
995338558 has a semiprime digit sum 55 in base 10.
995338558 has a Fibonacci digit sum 55 in base 10.
995338558 has a triangular digit sum 55 in base 10.
Reversing the decimal digits of 995338558 results in a semiprime.
995338558 is the sum of 2 positive triangular numbers.
995338558 is the difference of 2 positive pentagonal numbers in 2 ways.
995338558 = 237² + 442² + 31545² is the sum of 3 positive squares.
995338558² = 438154560² + 893711042² is the sum of 2 positive squares in 1 way.
995338558² is the sum of 3 positive squares.
995338558 is a proper divisor of 1321⁴¹⁹³⁰ - 1.
995338558 = '99533' + '8558' is the concatenation of 2 sphenic numbers.

Number of the day: 4843

Properties of the number 4843:

4843 is a cyclic number.
4843 = 29 × 167 is semiprime and squarefree.
4843 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 4648 totatives.
4843 has a prime digit sum 19 in base 10.
4843 = 2422² - 2421² = 98² - 69² is the difference of 2 nonnegative squares in 2 ways.
4843 is the sum of 2 positive triangular numbers.
4843 is the difference of 2 positive pentagonal numbers in 1 way.
4843 = 1² + 9² + 69² is the sum of 3 positive squares.
4843² = 3340² + 3507² is the sum of 2 positive squares in 1 way.
4843² is the sum of 3 positive squares.
4843 is a proper divisor of 1669¹⁴ - 1.
4843 = '4' + '843' is the concatenation of 2 semiprime numbers.
4843 is an emirpimes in (at least) the following bases: 2, 7, 8, 9, 12, 14, 17, 19, 25, 35, 36, 37, 38, 44, 48, 50, 54, 58, 59, 60, 65, 68, 70, 71, 73, 74, 79, 82, 84, 85, 88, 90, 93, 95, 96, and 99.
4843 is palindromic in (at least) the following bases: 26, 40, 47, and -44.
4843 in base 25 = 7ii and consists of only the digits '7' and 'i'.
4843 in base 26 = 747 and consists of only the digits '4' and '7'.
4843 in base 39 = 377 and consists of only the digits '3' and '7'.
4843 in base 40 = 313 and consists of only the digits '1' and '3'.
4843 in base 46 = 2DD and consists of only the digits '2' and 'D'.
4843 in base 47 = 292 and consists of only the digits '2' and '9'.

The number 4843 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A047801: Number of different values of i^2+j^2+k^2+l^2 for i,j,k,l in [ 0,n ].
A060527: A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of 8 musical tones: 8/7 16/11 5/4 4/3 3/2 8/5 11/8 7/4.
A066831: Numbers n such that sigma(n) divides sigma(phi(n)).
A067382: Numbers n such that sigma(phi(n))/sigma(n) = 2.
A218042: Numbers n such that Q(sqrt(n)) has class number 10.
A264422: T(n,k)=Number of (n+1)X(k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having directed index change 0,1 2,2 1,0 -1,2 -2,-1 or -1,-1.
A324170: Numbers whose multiset multisystem (A302242) is crossing.
A324324: MM-numbers of crossing set partitions.
A326497: Number of maximal sum-free and product-free subsets of {1..n}.
A337562: Number of pairwise coprime strict compositions of n, where a singleton is always considered coprime.

Number of the day: 877311

Joseph-Louis Lagrange was born on this day 285 years ago.

Karl Hermann Amandus Schwarz was born on this day 178 years ago.

Properties of the number 877311:

877311 = 3⁴ × 10831 is the 807686^th composite number and is not squarefree.
877311 has 2 distinct prime factors, 10 divisors, 25 antidivisors and 584820 totatives.
877311 has a triangular digit product 1176 in base 10.
877311 = 34³ + 39³ + 92³ is the sum of 3 positive cubes in 1 way.
877311 = 438656² - 438655² = 146220² - 146217² = 48744² - 48735² = 16260² - 16233² = 5456² - 5375² is the difference of 2 nonnegative squares in 5 ways.
877311 is the sum of 2 positive triangular numbers.
877311 is the difference of 2 positive pentagonal numbers in 1 way.
877311 is not the sum of 3 positive squares.
877311² is the sum of 3 positive squares.
877311 is a proper divisor of 107¹¹⁴ - 1.
877311 = '877' + '311' is the concatenation of 2 prime numbers.
877311 = '87' + '7311' is the concatenation of 2 semiprime numbers.

Number of the day: 503062

Properties of the number 503062:

503062 = 2 × 7 × 35933 is a sphenic number and squarefree.
503062 has 3 distinct prime factors, 8 divisors, 17 antidivisors and 215592 totatives.
Reversing the decimal digits of 503062 results in a sphenic number.
503062 is the sum of 2 positive triangular numbers.
503062 is the difference of 2 positive pentagonal numbers in 4 ways.
503062 = 41² + 66² + 705² is the sum of 3 positive squares.
503062² = 270088² + 424410² is the sum of 2 positive squares in 1 way.
503062² is the sum of 3 positive squares.
503062 is a proper divisor of 379¹³⁸² - 1.
503062 is palindromic in (at least) base 82.

Number of the day: 10312

David Hilbert was born on this day 159 years ago.

Properties of the number 10312:

10312 is the 2443^th totient number.
10312 = 2³ × 1289 is the 9047^th composite number and is not squarefree.
10312 has 2 distinct prime factors, 8 divisors, 21 antidivisors and 5152 totatives.
10312 has a prime digit sum 7 in base 10.
Reversing the decimal digits of 10312 results in a sphenic number.
10312 = 2579² - 2577² = 1291² - 1287² is the difference of 2 nonnegative squares in 2 ways.
10312 is the difference of 2 positive pentagonal numbers in 2 ways.
10312 = 54² + 86² is the sum of 2 positive squares in 1 way.
10312 = 40² + 66² + 66² is the sum of 3 positive squares.
10312² = 4480² + 9288² is the sum of 2 positive squares in 1 way.
10312² is the sum of 3 positive squares.
10312 is a proper divisor of 479⁴ - 1.
10312 = '1031' + '2' is the concatenation of 2 prime numbers.
10312 is palindromic in (at least) the following bases: -26, and -61.
10312 in base 25 = gcc and consists of only the digits 'c' and 'g'.
10312 in base 58 = 33k and consists of only the digits '3' and 'k'.

The number 10312 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A064837: a(n) = (6*n^4 + 30*n^3 - 20*n^2 + 14)*n/30 + (n mod 2).
A087439: Expansion of (1-4x)/((1-x)(1-3x)(1-5x)).
A089396: Smallest n-digit member of A089395.
A116042: n+phi(n)+phi(phi(n)) is a cube.
A157159: Infinite product representation of series 1 - log(1-x)= 1 + sum((j-1)!*(x^j)/j!, j=1..infinity).
A207625: Triangle of coefficients of polynomials u(n,x) jointly generated with A207626; see the Formula section.
A215746: Numerator of sum(i=0..n, (-1)^i*4/(2*i + 1)).
A290892: p-INVERT of the positive integers, where p(S) = 1 - S^4.
A291817: G.f. A(x) satisfies: A(x - 3*x*A(x)) = x + x*A(x).
A326472: Sum of the second largest parts of the partitions of n into 9 parts.

Number of the day: 48502358637

Michel Loève was born on this day 114 years ago.

Properties of the number 48502358637:

48502358637 = 3 × 41² × 9617759 is composite and not squarefree.
48502358637 has 3 distinct prime factors, 12 divisors, 27 antidivisors and 31546246240 totatives.
48502358637 has an emirpimes digit sum 51 in base 10.
48502358637 = 24251179319² - 24251179318² = 8083726441² - 8083726438² = 591492199² - 591492158² = 197164121² - 197163998² = 14427479² - 14425798² = 4811401² - 4806358² is the difference of 2 nonnegative squares in 6 ways.
48502358637 is the difference of 2 positive pentagonal numbers in 3 ways.
48502358637 = 82² + 467² + 220232² is the sum of 3 positive squares.
48502358637² = 10646859213² + 47319374280² = 20774359440² + 43828127763² is the sum of 2 positive squares in 2 ways.
48502358637² is the sum of 3 positive squares.
48502358637 is a proper divisor of 1229^3010138 - 1.
48502358637 = '4' + '8502358637' is the concatenation of 2 semiprime numbers.

Number of the day: 920897

Properties of the number 920897:

920897 is a cyclic number.
920897 = 23 × 40039 is semiprime and squarefree.
920897 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 880836 totatives.
920897 has a semiprime digit sum 35 in base 10.
Reversing the decimal digits of 920897 results in an emirpimes.
920897 = 460449² - 460448² = 20031² - 20008² is the difference of 2 nonnegative squares in 2 ways.
920897 is the difference of 2 positive pentagonal numbers in 2 ways.
920897 = 20² + 81² + 956² is the sum of 3 positive squares.
920897² is the sum of 3 positive squares.
920897 is a proper divisor of 691⁶⁶⁷³ - 1.
920897 is an emirpimes in (at least) the following bases: 7, 9, 10, 13, 17, 18, 19, 20, 21, 23, 27, 29, 33, 36, 37, 40, 41, 46, 51, 59, 62, 64, 66, 69, 72, 76, 77, 79, 81, 84, 85, 86, 88, 89, 92, 94, 95, 96, and 97.

Number of the day: 632073

Properties of the number 632073:

632073 = 3 × 13 × 19 × 853 is the 580580^th composite number and is squarefree.
632073 has 4 distinct prime factors, 16 divisors, 19 antidivisors and 368064 totatives.
632073 has a semiprime digit sum 21 in base 10.
632073 has a Fibonacci digit sum 21 in base 10.
632073 has a triangular digit sum 21 in base 10.
632073 = 316037² - 316036² = 105347² - 105344² = 24317² - 24304² = 16643² - 16624² = 8123² - 8084² = 5573² - 5516² = 1403² - 1156² = 797² - 56² is the difference of 2 nonnegative squares in 8 ways.
632073 is the difference of 2 positive pentagonal numbers in 3 ways.
632073 = 26² + 31² + 794² is the sum of 3 positive squares.
632073² = 243105² + 583452² = 95760² + 624777² = 376200² + 507927² = 151905² + 613548² is the sum of 2 positive squares in 4 ways.
632073² is the sum of 3 positive squares.
632073 is a proper divisor of 1471¹² - 1.
632073 = '6' + '32073' is the concatenation of 2 semiprime numbers.

Number of the day: 208

Guido Fubini was born on this day 142 years ago.

Properties of the number 208:

208 is the 74^th totient number.
208 = 2⁴ × 13 is the 161^th composite number and is not squarefree.
208 has 2 distinct prime factors, 10 divisors, 5 antidivisors and 96 totatives.
208 has a semiprime digit sum 10 in base 10.
208 has a triangular digit sum 10 in base 10.
208 has sum of divisors equal to 434 which is a sphenic number.
Reversing the decimal digits of 208 results in a semiprime.
208 = 6³ - 2³ is the difference of 2 positive cubes in 1 way.
208 = 53² - 51² = 28² - 24² = 17² - 9² is the difference of 2 nonnegative squares in 3 ways.
208 is the sum of 2 positive triangular numbers.
208 is the difference of 2 positive pentagonal numbers in 1 way.
208 = 8² + 12² is the sum of 2 positive squares in 1 way.
208 is not the sum of 3 positive squares.
208² = 80² + 192² is the sum of 2 positive squares in 1 way.
208² is the sum of 3 positive squares.
208 is a proper divisor of 79² - 1.
208 is palindromic in (at least) the following bases: 15, 25, 51, -7, -23, and -69.
208 in base 5 = 1313 and consists of only the digits '1' and '3'.
208 in base 6 = 544 and consists of only the digits '4' and '5'.

The number 208 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001082: Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...
A001399: a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.
A005101: Abundant numbers (sum of divisors of n exceeds 2n).
A005153: Practical numbers: positive integers m such that every k <= sigma(m) is a sum of distinct divisors of m. Also called panarithmic numbers.
A008574: a(0) = 1, thereafter a(n) = 4n.
A008586: Multiples of 4.
A009766: Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).
A024450: Sum of squares of the first n primes.
A139251: First differences of toothpicks numbers A139250.
A270929: Numbers k such that (16*10^k - 31)/3 is prime.

Number of the day: 31690

Properties of the number 31690:

31690 = 2 × 5 × 3169 is a sphenic number and squarefree.
31690 has 3 distinct prime factors, 8 divisors, 11 antidivisors and 12672 totatives.
31690 has a prime digit sum 19 in base 10.
31690 = (46 × 47)/2 + … + (65 × 66)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
31690 is the sum of 2 positive triangular numbers.
31690 is the difference of 2 positive pentagonal numbers in 2 ways.
31690 = 91² + 153² = 19² + 177² is the sum of 2 positive squares in 2 ways.
31690 = 81² + 100² + 123² is the sum of 3 positive squares.
31690² = 19014² + 25352² = 15128² + 27846² = 6726² + 30968² = 13200² + 28810² is the sum of 2 positive squares in 4 ways.
31690² is the sum of 3 positive squares.
31690 is a proper divisor of 79¹² - 1.
31690 is palindromic in (at least) the following bases: 55, and 62.
31690 in base 32 = uua and consists of only the digits 'a' and 'u'.
31690 in base 44 = GGA and consists of only the digits 'A' and 'G'.
31690 in base 54 = Akk and consists of only the digits 'A' and 'k'.
31690 in base 55 = AQA and consists of only the digits 'A' and 'Q'.
31690 in base 61 = 8VV and consists of only the digits '8' and 'V'.
31690 in base 62 = 8F8 and consists of only the digits '8' and 'F'.

The number 31690 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A084569: Partial sums of A084570.
A085146: Numbers n such that n!!!!+1 is prime.
A102437: Let pi be an unrestricted partition of n with the summands written in binary notation. a(n) is the number of such partitions whose binary representation has an odd number of binary ones.
A307240: a(0) = 1; a(n) = Sum_{k=1..n} -lambda(k+1)*a(n-k), where lambda() is the Liouville function (A008836).
A338277: Greatest integer k such that its square root is less than or equal to the Sum_{j=0..n} sqrt(j).
A339068: Number of unlabeled series-reduced 2-connected graphs with n edges.

Number of the day: 2629

Properties of the number 2629:

2629 is a cyclic number.
2629 = 11 × 239 is semiprime and squarefree.
2629 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 2380 totatives.
2629 has a prime digit sum 19 in base 10.
Reversing the decimal digits of 2629 results in a sphenic number.
2629 = 6³ + 6³ + 13³ is the sum of 3 positive cubes in 1 way.
2629 = 1315² - 1314² = 125² - 114² is the difference of 2 nonnegative squares in 2 ways.
2629 is the sum of 2 positive triangular numbers.
2629 is the difference of 2 positive pentagonal numbers in 2 ways.
2629 = 17² + 24² + 42² is the sum of 3 positive squares.
2629² is the sum of 3 positive squares.
2629 is a proper divisor of 1913² - 1.
2629 = '262' + '9' is the concatenation of 2 semiprime numbers.
2629 is an emirpimes in (at least) the following bases: 4, 5, 8, 9, 11, 13, 16, 17, 20, 21, 23, 37, 39, 40, 42, 44, 48, 50, 51, 53, 57, 58, 59, 60, 64, 66, 67, 68, 74, 75, 81, 88, 90, 93, 95, and 96.
2629 is palindromic in (at least) the following bases: 19, 25, 26, -37, and -73.
2629 in base 19 = 757 and consists of only the digits '5' and '7'.
2629 in base 24 = 4dd and consists of only the digits '4' and 'd'.
2629 in base 25 = 454 and consists of only the digits '4' and '5'.
2629 in base 26 = 3n3 and consists of only the digits '3' and 'n'.
2629 in base 29 = 33j and consists of only the digits '3' and 'j'.
2629 in base 36 = 211 and consists of only the digits '1' and '2'.

The number 2629 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000328: Number of points of norm <= n^2 in square lattice.
A028342: Expansion of Product_{i>=1} (1 - x^i)^(-1/i); also of exp(Sum_{n>=1} (d(n)*x^n/n)) where d is number of divisors function.
A038764: a(n) = (9*n^2 + 3*n + 2)/2.
A074343: a(1) = 7; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A080014: Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=2, r=2, I={1}.
A084849: a(n) = 1 + n + 2*n^2.
A088137: Generalized Gaussian Fibonacci integers.
A088410: a(n) = A069543(n)/8.
A108050: Integers n such that 10^n+21 is prime.
A113747: Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 10 multiples of n-1, n-2, ..., 1, for n>=1.

Number of the day: 2254

Properties of the number 2254:

2254 = 2 × 7² × 23 is the 1918^th composite number and is not squarefree.
2254 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 924 totatives.
2254 has an emirp digit sum 13 in base 10.
2254 has a Fibonacci digit sum 13 in base 10.
2254 is the difference of 2 positive pentagonal numbers in 4 ways.
2254 = 2² + 15² + 45² is the sum of 3 positive squares.
2254² is the sum of 3 positive squares.
2254 is a proper divisor of 1471² - 1.
2254 is palindromic in (at least) the following bases: 48, 97, and -25.
2254 in base 33 = 22a and consists of only the digits '2' and 'a'.

The number 2254 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A003363: Numbers that are the sum of 7 positive 6th powers.
A005282: Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.
A035928: Numbers n such that BCR(n) = n, where BCR = binary-complement-and-reverse = take one's complement then reverse bit order.
A128084: Triangle, read by rows of n^2+1 terms, of coefficients of q in the q-analog of the even double factorials: T(n,k) = [q^k] Product_{j=1..n} (1-q^(2j))/(1-q) for n>0, with T(0,0)=1.
A161699: Number of reduced words of length n in the Weyl group B_6.
A217089: Numbers n such that (n^97-1)/(n-1) is prime.
A238707: Number T(n,k) of ballot sequences of length n having difference k between the multiplicities of the smallest and the largest value; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A241619: T(n,k)=Number of length n+2 0..k arrays with no consecutive three elements summing to more than k
A249553: Numbers n such that there are precisely 10 groups of order n.
A272184: Numbers n such that Bernoulli number B_{n} has denominator 282.

Number of the day: 864315

Sofia Kovalevskaya was born on this day 171 years ago.

Properties of the number 864315:

864315 = 3² × 5 × 19207 is the 795647^th composite number and is not squarefree.
864315 has 3 distinct prime factors, 12 divisors, 15 antidivisors and 460944 totatives.
864315 = 432158² - 432157² = 144054² - 144051² = 86434² - 86429² = 48022² - 48013² = 28818² - 28803² = 9626² - 9581² is the difference of 2 nonnegative squares in 6 ways.
864315 is the sum of 2 positive triangular numbers.
864315 is the difference of 2 positive pentagonal numbers in 2 ways.
864315 = 11² + 175² + 913² is the sum of 3 positive squares.
864315² = 518589² + 691452² is the sum of 2 positive squares in 1 way.
864315² is the sum of 3 positive squares.
864315 is a proper divisor of 661³³ - 1.
864315 = '86' + '4315' is the concatenation of 2 semiprime numbers.

The number 864315 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A249513: Expansion of -(4*x*sqrt(4*x^2+1)+8*x^2+1)/((2*x^2-1)*sqrt(4*x^2+1) +4*x^3+x).

Number of the day: 94414

Alfred Tarski was born on this day 120 years ago.

Properties of the number 94414:

94414 = 2 × 47207 is semiprime and squarefree.
94414 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 47206 totatives.
94414 has a semiprime digit sum 22 in base 10.
Reversing the decimal digits of 94414 results in an emirpimes.
94414 is the difference of 2 positive pentagonal numbers in 2 ways.
94414 = 18² + 97² + 291² is the sum of 3 positive squares.
94414² is the sum of 3 positive squares.
94414 is a proper divisor of 3²³⁶⁰³ - 1.
94414 = '9' + '4414' is the concatenation of 2 semiprime numbers.
94414 is an emirpimes in (at least) the following bases: 6, 8, 10, 13, 16, 17, 27, 29, 31, 37, 38, 42, 49, 50, 56, 57, 60, 65, 67, 69, 70, 78, 81, 84, 85, 86, 89, 90, 91, 97, and 98.
94414 is palindromic in (at least) the following bases: 52, 80, and -49.
94414 in base 48 = ekk and consists of only the digits 'e' and 'k'.
94414 in base 52 = YlY and consists of only the digits 'Y' and 'l'.

Number of the day: 7265

Properties of the number 7265:

7265 is a cyclic number.
7265 = 5 × 1453 is semiprime and squarefree.
7265 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 5808 totatives.
7265 has an oblong digit sum 20 in base 10.
7265 has an oblong digit product 420 in base 10.
Reversing the decimal digits of 7265 results in an emirpimes.
7265 = 3633² - 3632² = 729² - 724² is the difference of 2 nonnegative squares in 2 ways.
7265 is the sum of 2 positive triangular numbers.
7265 is the difference of 2 positive pentagonal numbers in 2 ways.
7265 = 44² + 73² = 32² + 79² is the sum of 2 positive squares in 2 ways.
7265 = 10² + 21² + 82² is the sum of 3 positive squares.
7265² = 5056² + 5217² = 4359² + 5812² = 3393² + 6424² = 1140² + 7175² is the sum of 2 positive squares in 4 ways.
7265² is the sum of 3 positive squares.
7265 is a proper divisor of 131¹¹ - 1.
7265 is an emirpimes in (at least) the following bases: 2, 4, 5, 10, 17, 20, 22, 23, 25, 28, 31, 36, 37, 39, 41, 51, 53, 65, 66, 69, 71, 72, 73, 74, 77, 78, 81, 83, 89, 91, 92, 93, 94, and 95.
7265 is palindromic in (at least) base -26.
7265 in base 11 = 5505 and consists of only the digits '0' and '5'.
7265 in base 25 = bff and consists of only the digits 'b' and 'f'.
7265 in base 42 = 44f and consists of only the digits '4' and 'f'.

The number 7265 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A041642: Numerators of continued fraction convergents to sqrt(340).
A132581: Number of antichains in the first n elements of the infinite Boolean lattice.
A164754: Number of n X 2 1..4 arrays with all 1's connected, all 2's connected, all 3's connected, all 4's connected, 1 in the upper left corner, 2 in the upper right corner, 3 in the lower left corner, 4 in the lower right corner, and with no element having more than 2 neighbors with the same value.
A218568: Number of partitions p of n such that max(p)-min(p) = 5.
A239030: T(n,k)=Number of nXk 0..2 arrays with no element equal to the sum of elements to its left or one plus the sum of the elements above it, modulo 3
A248351: Numbers k such that 10^k + 987654321 is prime.
A270278: Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 139", based on the 5-celled von Neumann neighborhood.
A293971: Number of sets of exactly nine nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.
A325347: Number of partitions of n having an integer median.
A330445: Expansion of e.g.f.: Sum_{k>=1} log(1 + (exp(x) - 1)^k)/k.

Number of the day: 60838

Properties of the number 60838:

60838 = 2 × 19 × 1601 is a sphenic number and squarefree.
60838 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 28800 totatives.
60838 has a semiprime digit sum 25 in base 10.
Reversing the decimal digits of 60838 results in a semiprime.
60838 = 8³ + 29³ + 33³ is the sum of 3 positive cubes in 1 way.
60838 is the difference of 2 positive pentagonal numbers in 4 ways.
60838 = 5² + 42² + 243² is the sum of 3 positive squares.
60838² = 3040² + 60762² is the sum of 2 positive squares in 1 way.
60838² is the sum of 3 positive squares.
60838 is a proper divisor of 1291⁸ - 1.
60838 is palindromic in (at least) the following bases: 40, -40, -57, and -62.
60838 in base 40 = c0c and consists of only the digits '0' and 'c'.
60838 in base 56 = JMM and consists of only the digits 'J' and 'M'.
60838 in base 61 = GLL and consists of only the digits 'G' and 'L'.

The number 60838 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A163318: Expansion of g.f.: Product_{k>=1} 1+k*x^k/(1-x^k)^2.

Number of the day: 5178

Properties of the number 5178:

5178 is the 1302^th totient number.
5178 = 2 × 3 × 863 is a sphenic number and squarefree.
5178 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 1724 totatives.
5178 has a semiprime digit sum 21 in base 10.
5178 has a Fibonacci digit sum 21 in base 10.
5178 has a triangular digit sum 21 in base 10.
5178 is the difference of 2 positive pentagonal numbers in 2 ways.
5178 = 4² + 11² + 71² is the sum of 3 positive squares.
5178² is the sum of 3 positive squares.
5178 is a proper divisor of 19⁴³¹ - 1.
5178 is palindromic in (at least) the following bases: -21, and -45.
5178 in base 20 = cii and consists of only the digits 'c' and 'i'.
5178 in base 41 = 33C and consists of only the digits '3' and 'C'.

The number 5178 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A187206: a(n) = 6*(24*n - 1).
A191455: Dispersion of (floor(n*e)), by antidiagonals.
A218510: Number of partitions of n in which any two parts differ by at most 8.
A227364: a(n) = 1 + 2*3 + 4*5*6 + 7*8*9*10 + ... + ...*n(see Example lines).
A252210: T(n,k)=Number of (n+2)X(k+2) 0..3 arrays with every 3X3 subblock row and column sum equal to 1 3 4 6 or 7 and every 3X3 diagonal and antidiagonal sum not equal to 1 3 4 6 or 7
A257211: Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 8 as largest digit.
A268639: T(n,k)=Number of nXk 0..2 arrays with some element plus some horizontally or vertically adjacent neighbor totalling two exactly once.
A308872: Sum of the second largest parts in the partitions of n into 6 parts.
A308990: Sum of the smallest parts in the partitions of n into 8 parts.
A309619: a(n) = Sum_{k=0..floor(n/2)} k! * (n - 2*k)!.

Number of the day: 5370249304850

Issai Schur was born on this day 146 years ago.

Properties of the number 5370249304850:

5370249304850 = 2 × 5² × 61 × 11579 × 152063 is composite and not squarefree.
5370249304850 has 5 distinct prime factors, 48 divisors, 35 antidivisors and 2112688603200 totatives.
5370249304850 is the difference of 2 positive pentagonal numbers in 11 ways.
5370249304850 = 64² + 3625² + 2317377² is the sum of 3 positive squares.
5370249304850² = 3222149582910² + 4296199443880² = 2394602968720² + 4806813312210² = 3644726577390² + 3944051948480² = 968405612350² + 5282212431000² = 1503669805358² + 5155439332656² = 549350092824² + 5342077505218² = 2408688868536² + 4799770362302² is the sum of 2 positive squares in 7 ways.
5370249304850² is the sum of 3 positive squares.
5370249304850 is a proper divisor of 499^251510548 - 1.

Number of the day: 7763

Properties of the number 7763:

7763 is a cyclic number.
7763 = 7 × 1109 is semiprime and squarefree.
7763 has 2 distinct prime factors, 4 divisors, 25 antidivisors and 6648 totatives.
7763 has a prime digit sum 23 in base 10.
Reversing the decimal digits of 7763 results in a prime.
7763 = 3882² - 3881² = 558² - 551² is the difference of 2 nonnegative squares in 2 ways.
7763 is the difference of 2 positive pentagonal numbers in 2 ways.
7763 = 5² + 13² + 87² is the sum of 3 positive squares.
7763² = 987² + 7700² is the sum of 2 positive squares in 1 way.
7763² is the sum of 3 positive squares.
7763 is a proper divisor of 29²⁷⁷ - 1.
7763 is an emirpimes in (at least) the following bases: 3, 6, 9, 11, 13, 15, 16, 20, 21, 23, 24, 31, 32, 33, 35, 37, 46, 50, 51, 53, 55, 56, 59, 63, 65, 67, 69, 70, 71, 72, 73, 74, 77, 78, 79, 80, 81, 82, 86, 95, 97, and 100.
7763 is palindromic in (at least) the following bases: -25, and -33.
7763 in base 6 = 55535 and consists of only the digits '3' and '5'.
7763 in base 24 = dbb and consists of only the digits 'b' and 'd'.

The number 7763 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002597: Number of partitions into one kind of 1's, two kinds of 2's, and three kinds of 3's.
A034134: Decimal part of cube root of a(n) starts with 8: first term of runs.
A116021: phi(n) plus the n-th prime gives a square.
A157036: Shorthand for A157035, the largest prime with 2^n digits.
A227954: Smallest m such that A070965(m) = -n.
A234696: Indices of primes in the tribonacci-like sequence, A214727.
A253054: If, for some m, A098550(m-2) is a prime p and A098550(m) = 7p, add 7p to the sequence.
A272989: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 566", based on the 5-celled von Neumann neighborhood.
A275545: Number of new duplicate terms at a given iteration of the Collatz (or 3x+1) map starting with 0.
A278920: In the binary race of Pi, where the race leader changes.

Number of the day: 535066

Richard Courant was born on this day 133 years ago.

Properties of the number 535066:

535066 = 2 × 7 × 38219 is a sphenic number and squarefree.
535066 has 3 distinct prime factors, 8 divisors, 9 antidivisors and 229308 totatives.
535066 has a semiprime digit sum 25 in base 10.
535066 is the difference of 2 positive pentagonal numbers in 4 ways.
535066 = 171² + 184² + 687² is the sum of 3 positive squares.
535066² is the sum of 3 positive squares.
535066 is a proper divisor of 859⁵⁸² - 1.
535066 is palindromic in (at least) base 96.

Number of the day: 926906

Émile Borel was born on this day 150 years ago.

Properties of the number 926906:

926906 = 2 × 463453 is semiprime and squarefree.
926906 has 2 distinct prime factors, 4 divisors, 25 antidivisors and 463452 totatives.
Reversing the decimal digits of 926906 results in an emirpimes.
926906 = 85² + 959² is the sum of 2 positive squares in 1 way.
926906 = 56² + 89² + 957² is the sum of 3 positive squares.
926906² = 163030² + 912456² is the sum of 2 positive squares in 1 way.
926906² is the sum of 3 positive squares.
926906 is a proper divisor of 71³⁵¹¹ - 1.
926906 is an emirpimes in (at least) the following bases: 2, 4, 10, 13, 19, 22, 24, 28, 34, 36, 40, 46, 50, 52, 53, 54, 58, 62, 65, 71, 77, 79, 84, 95, and 96.

Number of the day: 259236

Properties of the number 259236:

259236 = 2² × 3² × 19 × 379 is the 236455^th composite number and is not squarefree.
259236 has 4 distinct prime factors, 36 divisors, 11 antidivisors and 81648 totatives.
259236 has a triangular digit product 3240 in base 10.
259236 = 64810² - 64808² = 21606² - 21600² = 7210² - 7192² = 3430² - 3392² = 1194² - 1080² = 550² - 208² is the difference of 2 nonnegative squares in 6 ways.
259236 is the difference of 2 positive pentagonal numbers in 2 ways.
259236 = 4² + 34² + 508² is the sum of 3 positive squares.
259236² is the sum of 3 positive squares.
259236 is a proper divisor of 809⁶ - 1.
259236 = '25923' + '6' is the concatenation of 2 semiprime numbers.
259236 is palindromic in (at least) the following bases: 78, and 98.

The number 259236 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A136237: Matrix cube of triangle V = A136230, read by rows.
A257949: The q-series expansion of an expression related to fermionic systems.

Number of the day: 804682

Camille Jordan was born on this day 183 years ago.

Properties of the number 804682:

804682 = 2 × 402341 is semiprime and squarefree.
804682 has 2 distinct prime factors, 4 divisors, 21 antidivisors and 402340 totatives.
804682 has a triangular digit sum 28 in base 10.
804682 is the sum of 2 positive triangular numbers.
804682 is the difference of 2 positive pentagonal numbers in 2 ways.
804682 = 179² + 879² is the sum of 2 positive squares in 1 way.
804682 = 3² + 8² + 897² is the sum of 3 positive squares.
804682² = 314682² + 740600² is the sum of 2 positive squares in 1 way.
804682² is the sum of 3 positive squares.
804682 is a proper divisor of 37²⁰¹¹⁷ - 1.
804682 is an emirpimes in (at least) the following bases: 4, 5, 14, 15, 19, 20, 22, 23, 30, 31, 35, 36, 43, 44, 45, 55, 65, 66, 67, 70, 71, 74, 82, 84, 85, 86, 90, 92, 94, and 98.

Number of the day: 109836

Sir Isaac Newton was born on this day 378 years ago.

Properties of the number 109836:

109836 = 2² × 3⁵ × 113 is the 99398^th composite number and is not squarefree.
109836 has 3 distinct prime factors, 36 divisors, 15 antidivisors and 36288 totatives.
109836 = 27460² - 27458² = 9156² - 9150² = 3060² - 3042² = 1044² - 990² = 420² - 258² = 356² - 130² is the difference of 2 nonnegative squares in 6 ways.
109836 is the sum of 2 positive triangular numbers.
109836 = 26² + 74² + 322² is the sum of 3 positive squares.
109836² = 14580² + 108864² is the sum of 2 positive squares in 1 way.
109836² is the sum of 3 positive squares.
109836 is a proper divisor of 487¹⁶ - 1.
109836 is palindromic in (at least) the following bases: 63, 74, 86, -85, and -96.
109836 in base 26 = 66cc and consists of only the digits '6' and 'c'.
109836 in base 60 = UUa and consists of only the digits 'U' and 'a'.

Number of the day: 623

Properties of the number 623:

623 is a cyclic number.
623 = 7 × 89 is semiprime and squarefree.
623 has 2 distinct prime factors, 4 divisors, 11 antidivisors and 528 totatives.
623 has a prime digit sum 11 in base 10.
623 has a triangular digit product 36 in base 10.
Reversing the decimal digits of 623 results in an emirpimes.
623 = 4³ + 6³ + 7³ is the sum of 3 positive cubes in 1 way.
623 = 312² - 311² = 48² - 41² is the difference of 2 nonnegative squares in 2 ways.
623 is the sum of 2 positive triangular numbers.
623 is the difference of 2 positive pentagonal numbers in 2 ways.
623 is not the sum of 3 positive squares.
623² = 273² + 560² is the sum of 2 positive squares in 1 way.
623² is the sum of 3 positive squares.
623 is a proper divisor of 179³ - 1.
623 is an emirpimes in (at least) the following bases: 2, 4, 10, 14, 15, 17, 19, 22, 24, 27, 28, 29, 36, 38, 39, 43, 49, 51, 54, 56, 57, 59, 61, 64, 65, 66, 67, 73, 74, 78, 83, 84, 93, 95, 97, and 100.
623 is palindromic in (at least) the following bases: 88, and -23.
623 in base 5 = 4443 and consists of only the digits '3' and '4'.
623 in base 17 = 22b and consists of only the digits '2' and 'b'.
623 in base 24 = 11n and consists of only the digits '1' and 'n'.

The number 623 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000701: One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.
A008865: a(n) = n^2 - 2.
A014616: a(n) = solution to the postage stamp problem with 2 denominations and n stamps.
A027187: Number of partitions of n into an even number of parts.
A036236: Least inverse of A015910: smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists.
A038622: Triangular array that counts rooted polyominoes.
A127820: a(n) = least k such that the remainder when 12^k is divided by k is n.
A183010: a(n) = 24*n - 1.
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.
A316652: Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Number of the day: 7207

Properties of the number 7207:

7207 is a cyclic number.
7207 is the 920^th prime.
7207 has 17 antidivisors and 7206 totatives.
Reversing the decimal digits of 7207 results in an emirp.
7207 = 3604² - 3603² is the difference of 2 nonnegative squares in 1 way.
7207 is the difference of 2 positive pentagonal numbers in 1 way.
7207 is not the sum of 3 positive squares.
7207² is the sum of 3 positive squares.
7207 is a proper divisor of 11¹²⁰¹ - 1.
7207 is an emirp in (at least) the following bases: 5, 6, 10, 13, 17, 18, 19, 23, 31, 37, 39, 43, 44, 47, 51, 53, 59, 66, 67, 74, 76, 77, 78, 79, 84, 85, 89, and 97.
7207 is palindromic in (at least) the following bases: 32, and 55.
7207 in base 24 = cc7 and consists of only the digits '7' and 'c'.
7207 in base 31 = 7ff and consists of only the digits '7' and 'f'.
7207 in base 32 = 717 and consists of only the digits '1' and '7'.
7207 in base 54 = 2PP and consists of only the digits '2' and 'P'.
7207 in base 55 = 2L2 and consists of only the digits '2' and 'L'.

The number 7207 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A045713: Primes with first digit 7.
A062325: Numbers k for which phi(prime(k)) is a square.
A072481: a(n) = Sum_{k=1..n} Sum_{d=1..k} (k mod d).
A074340: a(1) = 5; a(n) is smallest number > a(n-1) such that the juxtaposition a(1)a(2)...a(n) is a prime.
A078850: Initial term in sequence of four consecutive primes separated by 3 consecutive differences each <=6 (i.e., when d=2,4 or 6) and forming d-pattern=[4,2,6]; short d-string notation of pattern = [426].
A103810: Primes from merging of 4 successive digits in decimal expansion of the Golden Ratio, (1+sqrt(5))/2.
A134971: Canyon primes.
A142019: Primes congruent to 15 mod 31.
A162622: Triangle read by rows in which row n lists n+1 terms, starting with n, such that the difference between successive terms is equal to n^4 - 1.
A162624: Triangle read by rows in which row n lists n terms, starting with n^4 + n - 1, such that the difference between successive terms is equal to n^4 - 1 = A123865(n).

Number of the day: 2021

Happy New Year!

Properties of the number 2021:

2021 is a cyclic number.
2021 = 43 × 47 is semiprime and squarefree.
2021 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 1932 totatives.
2021 has a prime digit sum 5 in base 10.
2021 has a Fibonacci digit sum 5 in base 10.
Reversing the decimal digits of 2021 results in an emirpimes.
2021 = 1011² - 1010² = 45² - 2² is the difference of 2 nonnegative squares in 2 ways.
2021 is the difference of 2 positive pentagonal numbers in 1 way.
2021 = 1² + 16² + 42² is the sum of 3 positive squares.
2021² is the sum of 3 positive squares.
2021 is a proper divisor of 1033² - 1.
2021 is an emirpimes in (at least) the following bases: 2, 4, 5, 10, 12, 14, 17, 21, 23, 25, 27, 29, 33, 36, 38, 39, 42, 44, 49, 51, 55, 57, 62, 67, 76, 81, 85, 86, 87, 89, 91, 94, 95, 96, and 98.
2021 is palindromic in (at least) the following bases: 46, -10, and -21.
2021 in base 20 = 511 and consists of only the digits '1' and '5'.
2021 in base 44 = 11f and consists of only the digits '1' and 'f'.

The number 2021 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001704: a(n) = n concatenated with n + 1.
A006094: Products of 2 successive primes.
A028347: a(n) = n^2 - 4.
A052219: Numbers whose sum of digits is 5.
A061037: Numerator of 1/4 - 1/n^2.
A078371: a(n) = (2*n+5)*(2*n+1).
A202018: a(n) = n^2 + n + 41.
A299258: Coordination sequence for 3D uniform tiling formed by stacking parallel layers of the 4.6.12 2D tiling (cf. A072154).
A299276: Partial sums of A008137.
A326256: MM-numbers of nesting multiset partitions.