Number of the day: 503135

Carl Friedrich Gauss was born on this day 240 years ago.

Claude Shannon was born on this day 101 years ago.

Properties of the number 503135:

503135 = 5 × 47 × 2141 is a sphenic number and squarefree.
503135 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 393760 totatives.
503135 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 503135 results in a semiprime.
503135 = 251568² - 251567² = 50316² - 50311² = 5376² - 5329² = 1188² - 953² is the difference of 2 nonnegative squares in 4 ways.
503135 is the difference of 2 positive pentagonal numbers in 4 ways.
503135 is not the sum of 3 positive squares.
503135² = 328248² + 381311² = 301881² + 402508² = 208351² + 457968² = 108100² + 491385² is the sum of 2 positive squares in 4 ways.
503135² is the sum of 3 positive squares.
503135 is a divisor of 941²⁰ - 1.

Number of the day: 841074

Henri Poincaré was born on this day 163 years ago.

Properties of the number 841074:

841074 = 2 × 3 × 13 × 41 × 263 is the 774107^th composite number and is squarefree.
841074 has 5 distinct prime factors, 32 divisors, 27 antidivisors and 251520 totatives.
841074 is the sum of 2 positive triangular numbers.
841074 is the difference of 2 positive pentagonal numbers in 2 ways.
841074 = 32² + 223² + 889² is the sum of 3 positive squares.
841074² = 184626² + 820560² = 145176² + 828450² = 486024² + 686430² = 323490² + 776376² is the sum of 2 positive squares in 4 ways.
841074² is the sum of 3 positive squares.
841074 is a divisor of 1579⁶⁰ - 1.
841074 = '84107' + '4' is the concatenation of 2 semiprime numbers.

Number of the day: 74002

Kurt Gödel was born on this day 111 years ago.

Properties of the number 74002:

74002 = 2 × 163 × 227 is a sphenic number and squarefree.
74002 has 3 distinct prime factors, 8 divisors, 53 antidivisors and 36612 totatives.
74002 has an emirp digit sum 13 in base 10.
74002 has a Fibonacci digit sum 13 in base 10.
Reversing the decimal digits of 74002 results in a prime.
74002 is the difference of 2 positive pentagonal numbers in 3 ways.
74002 = 3² + 52² + 267² is the sum of 3 positive squares.
74002² is the sum of 3 positive squares.
74002 is a divisor of 907¹⁶² - 1.
74002 is palindromic in (at least) the following bases: 44, 46, 64, and 66.
74002 in base 16 = 12112 and consists of only the digits '1' and '2'.
74002 in base 17 = f111 and consists of only the digits '1' and 'f'.
74002 in base 44 = c9c and consists of only the digits '9' and 'c'.
74002 in base 46 = YiY and consists of only the digits 'Y' and 'i'.

The number 74002 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A032936: Numbers whose set of base 16 digits is {1,2}.

Number of the day: 74981

Properties of the number 74981:

74981 = 97 × 773 is semiprime and squarefree.
74981 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 74112 totatives.
74981 has a prime digit sum 29 in base 10.
74981 has a triangular digit product 2016 in base 10.
Reversing the decimal digits of 74981 results in a prime.
74981 = 37491² - 37490² = 435² - 338² is the difference of 2 nonnegative squares in 2 ways.
74981 is the difference of 2 positive pentagonal numbers in 2 ways.
74981 = 130² + 241² = 65² + 266² is the sum of 2 positive squares in 2 ways.
74981 = 6² + 31² + 272² is the sum of 3 positive squares.
74981² = 50245² + 55656² = 34580² + 66531² = 41181² + 62660² = 18915² + 72556² is the sum of 2 positive squares in 4 ways.
74981² is the sum of 3 positive squares.
74981 is a divisor of 1229⁴⁸ - 1.
74981 is an emirpimes in (at least) the following bases: 5, 13, 14, 15, 16, 19, 20, 34, 35, 36, 41, 43, 44, 45, 46, 47, 52, 54, 61, 67, 69, 71, 74, 75, 80, 81, 83, 88, 89, 91, 93, 99, and 100.
74981 is palindromic in (at least) the following bases: -20, and -85.
74981 in base 49 = VBB and consists of only the digits 'B' and 'V'.

Number of the day: 63348

Properties of the number 63348:

63348 = 2² × 3 × 5279 is the 57000^th composite number and is not squarefree.
63348 has 3 distinct prime factors, 12 divisors, 11 antidivisors and 21112 totatives.
63348 has sum of divisors equal to 147840 which is an oblong number.
63348 = 15838² - 15836² = 5282² - 5276² is the difference of 2 nonnegative squares in 2 ways.
63348 is the difference of 2 positive pentagonal numbers in 1 way.
63348 = 20² + 38² + 248² is the sum of 3 positive squares.
63348² is the sum of 3 positive squares.
63348 is a divisor of 541¹⁴ - 1.
63348 is palindromic in (at least) the following bases: 44, and -97.
63348 in base 44 = WVW and consists of only the digits 'V' and 'W'.

The number 63348 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A054209: Consider all integer triples (i,j,k), j >= k>0, with i^3=binomial(j+2,3)+binomial(k+2,3), ordered by increasing i; sequence gives j values.
A126411: Number of base 24 n-digit numbers with adjacent digits differing by two or less.

Number of the day: 45567089187007

Felix Klein was born on this day 168 years ago.

Andrey Kolmogorov was born on this day 114 years ago.

Properties of the number 45567089187007:

45567089187007 = 29² × 2351 × 3361 × 6857 is composite and not squarefree.
45567089187007 has 4 distinct prime factors, 24 divisors, 117 antidivisors and 43957600512000 totatives.
45567089187007 has a prime digit sum 67 in base 10.
45567089187007 is the difference of 2 nonnegative squares in 12 ways.
45567089187007 is the difference of 2 positive pentagonal numbers in 11 ways.
45567089187007 is not the sum of 3 positive squares.
45567089187007² = 22776765794160² + 39466169777857² = 19326696016575² + 41265463017632² = 26060751527745² + 37379096388032² = 12870836187993² + 43711568180500² = 9094689434268² + 44650265856665² = 16553131139268² + 42454133679335² = 10724528064820² + 44287064869593² = 6907108471655² + 45040553610468² = 14463746344985² + 43210642886268² = 20825648402143² + 40529643294840² = 17291962746968² + 42158601036255² = 24207477207368² + 38605150747425² = 24673717274657² + 38308840922520² = 21315468059032² + 40274190723105² = 27852050220232² + 36064150003935² = 31425578749660² + 32996857687143² = 28495900870745² + 35557604679468² = 30195503594868² + 34126107009575² = 2221463325407² + 45512907154680² = 1669536713768² + 45536493762015² = 6096264857032² + 45157448685345² = 3887523286335² + 45400955713232² is the sum of 2 positive squares in 22 ways.
45567089187007² is the sum of 3 positive squares.
45567089187007 is a divisor of 137^11278120 - 1.
45567089187007 = '455' + '67089187007' is the concatenation of 2 sphenic numbers.

Number of the day: 4230

Properties of the number 4230:

4230 = 2 × 3² × 5 × 47 is the 3650^th composite number and is not squarefree.
4230 has 4 distinct prime factors, 24 divisors, 13 antidivisors and 1104 totatives.
4230 has a semiprime digit sum 9 in base 10.
4230 = 31² + … + 34² is the sum of at least 2 consecutive positive squares in 1 way.
4230 is the sum of 2 positive triangular numbers.
4230 is the difference of 2 positive pentagonal numbers in 1 way.
4230 = 1² + 2² + 65² is the sum of 3 positive squares.
4230² = 2538² + 3384² is the sum of 2 positive squares in 1 way.
4230² is the sum of 3 positive squares.
4230 is a divisor of 1223⁴ - 1.
4230 = '42' + '30' is the concatenation of 2 sphenic numbers.
4230 = '42' + '30' is the concatenation of 2 oblong numbers.
4230 is palindromic in (at least) the following bases: 17, 20, 21, 89, 93, and -19.
4230 in base 17 = eae and consists of only the digits 'a' and 'e'.
4230 in base 20 = aba and consists of only the digits 'a' and 'b'.
4230 in base 21 = 9c9 and consists of only the digits '9' and 'c'.
4230 in base 26 = 66i and consists of only the digits '6' and 'i'.
4230 in base 32 = 446 and consists of only the digits '4' and '6'.
4230 in base 37 = 33C and consists of only the digits '3' and 'C'.

The number 4230 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000500: Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.
A027575: a(n) = n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2.
A054000: a(n) = 2*n^2 - 2.
A125014: Numbers n for which nontrivial positive magic squares of exactly 7 different orders with magic sum n exist. For a definition of nontrivial positive magic squares, see A125005.
A126011: A106486-encodings for the minimal representatives of each equivalence class of the finite combinatorial games.
A135191: Numbers n that raised to the powers from 1 to k (with k>=1) are multiple of the sum of their digits (n raised to k+1 must not be a multiple). Case k=6.
A201596: Record (maximal) gaps between prime triplets (p, p+4, p+6).
A210095: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays containing all values 0..2 with every 2X2 subblock having one or two distinct values, and new values 0..2 introduced in row major order
A230899: T(n,k)=Number of nXk 0..3 arrays x(i,j) with each element horizontally or vertically next to at least one element with value (x(i,j)+1) mod 4 and at least one element with value (x(i,j)-1) mod 4, no adjacent elements equal, and upper left element zero
A256753: Numbers n such that n is both the average of some twin prime pair p, q (q = p+2) (i.e., n = p+1 = q-1) and is also the average of the prime before p and the prime after q.

Number of the day: 378766

Properties of the number 378766:

378766 = 2 × 229 × 827 is a sphenic number and squarefree.
378766 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 188328 totatives.
378766 has an emirp digit sum 37 in base 10.
Reversing the decimal digits of 378766 results in a semiprime.
378766 is the sum of 2 positive triangular numbers.
378766 is the difference of 2 positive pentagonal numbers in 3 ways.
378766 = 10² + 21² + 615² is the sum of 3 positive squares.
378766² = 99240² + 365534² is the sum of 2 positive squares in 1 way.
378766² is the sum of 3 positive squares.
378766 is a divisor of 1373¹¹⁸ - 1.
378766 is palindromic in (at least) base 90.

Number of the day: 734936

Properties of the number 734936:

734936 = 2³ × 91867 is the 675786^th composite number and is not squarefree.
734936 has 2 distinct prime factors, 8 divisors, 25 antidivisors and 367464 totatives.
Reversing the decimal digits of 734936 results in a semiprime.
734936 = 183735² - 183733² = 91869² - 91865² is the difference of 2 nonnegative squares in 2 ways.
734936 = 60² + 94² + 850² is the sum of 3 positive squares.
734936² is the sum of 3 positive squares.
734936 is a divisor of 1097³⁶⁶ - 1.
734936 = '73493' + '6' is the concatenation of 2 semiprime numbers.

Number of the day: 40603

Properties of the number 40603:

40603 = 19 × 2137 is semiprime and squarefree.
40603 has 2 distinct prime factors, 4 divisors, 19 antidivisors and 38448 totatives.
40603 has an emirp digit sum 13 in base 10.
40603 has a Fibonacci digit sum 13 in base 10.
40603 = 20302² - 20301² = 1078² - 1059² is the difference of 2 nonnegative squares in 2 ways.
40603 is the sum of 2 positive triangular numbers.
40603 is the difference of 2 positive pentagonal numbers in 2 ways.
40603 = 9² + 11² + 201² is the sum of 3 positive squares.
40603² = 8645² + 39672² is the sum of 2 positive squares in 1 way.
40603² is the sum of 3 positive squares.
40603 is a divisor of 37¹⁷⁸ - 1.
40603 is an emirpimes in (at least) the following bases: 4, 7, 12, 14, 23, 25, 26, 28, 30, 31, 32, 34, 35, 42, 47, 48, 52, 64, 65, 67, 71, 72, 75, 77, 79, 80, 88, 95, 97, and 99.
40603 is palindromic in (at least) the following bases: 36, 55, and 59.
40603 in base 6 = 511551 and consists of only the digits '1' and '5'.
40603 in base 36 = vbv and consists of only the digits 'b' and 'v'.
40603 in base 54 = Dnn and consists of only the digits 'D' and 'n'.
40603 in base 55 = DND and consists of only the digits 'D' and 'N'.
40603 in base 59 = BdB and consists of only the digits 'B' and 'd'.

The number 40603 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A075294: Interprimes which are of the form s*prime, s=19.
A100150: Structured snub cubic numbers.
A151479: Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0), ending on the vertical axis and consisting of n steps taken from {(-1, -1), (-1, 1), (0, -1), (0, 1), (1, 0), (1, 1)}
A211549: Number of (n+1)X(n+1) -9..9 symmetric matrices with every 2X2 subblock having sum zero and one or two distinct values
A213823: Principal diagonal of the convolution array A213822.

Number of the day: 4353

Properties of the number 4353:

4353 = 3 × 1451 is semiprime and squarefree.
4353 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 2900 totatives.
4353 has an emirpimes digit sum 15 in base 10.
4353 has a triangular digit sum 15 in base 10.
4353 = 2177² - 2176² = 727² - 724² is the difference of 2 nonnegative squares in 2 ways.
4353 is the difference of 2 positive pentagonal numbers in 1 way.
4353 = 1² + 16² + 64² is the sum of 3 positive squares.
4353² is the sum of 3 positive squares.
4353 is a divisor of 1021⁵ - 1.
4353 = '43' + '53' is the concatenation of 2 prime numbers.
4353 = '435' + '3' is the concatenation of 2 triangular numbers.
4353 is an emirpimes in (at least) the following bases: 17, 19, 25, 28, 33, 35, 36, 37, 39, 43, 45, 51, 54, 60, 62, 63, 65, 66, 71, 79, and 84.
4353 is palindromic in (at least) the following bases: 8, 64, -8, -27, and -68.
4353 in base 4 = 1010001 and consists of only the digits '0' and '1'.
4353 in base 16 = 1101 and consists of only the digits '0' and '1'.
4353 in base 17 = f11 and consists of only the digits '1' and 'f'.
4353 in base 26 = 6bb and consists of only the digits '6' and 'b'.
4353 in base 29 = 553 and consists of only the digits '3' and '5'.
4353 in base 46 = 22T and consists of only the digits '2' and 'T'.

The number 4353 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005744: G.f.: x*(1+x-x^2)/((1-x)^4*(1+x)).
A029705: Squarefree n such that Q(sqrt(n)) has class number 5.
A033052: a(1) = 1, a(2n) = 16a(n), a(2n+1) = a(2n)+1.
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.
A114166: Numbers n such that p(5n) is prime, where p(n) is the number of partitions of n.
A135110: Positive numbers such that the digital sum base 2 and the digital sum base 10 are in a ratio of 2:10.
A186651: Total number of positive integers below 10^n requiring 3 positive biquadrates in their representation as sum of biquadrates.
A195146: Concentric 16-gonal numbers.
A254204: T(n,k)=Number of length n 1..(k+2) arrays with no leading or trailing partial sum equal to a prime
A269494: T(n,k)=Number of length-n 0..k arrays with no repeated value differing from the previous repeated value by one.

Number of the day: 983

Properties of the number 983:

983 is the 166^th prime.
983 has 9 antidivisors and 982 totatives.
983 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 983 results in an emirp.
983 = 492² - 491² is the difference of 2 nonnegative squares in 1 way.
983 is the difference of 2 positive pentagonal numbers in 1 way.
983 is not the sum of 3 positive squares.
983² is the sum of 3 positive squares.
983 is a divisor of 2⁴⁹¹ - 1.
983 is an emirp in (at least) the following bases: 3, 4, 5, 8, 9, 10, 11, 13, 14, 15, 16, 19, 21, 23, 24, 25, 27, 28, 31, 33, 38, 39, 41, 43, 49, 50, 51, 55, 56, 61, 62, 64, 69, 71, 74, 75, 79, 82, 83, 84, 85, 87, 89, and 91.
983 is palindromic in (at least) base -20.
983 in base 4 = 33113 and consists of only the digits '1' and '3'.

The number 983 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A000057: Primes dividing all Fibonacci sequences.
A001190: Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n-1 nodes in all).
A001913: Full reptend primes: primes with primitive root 10.
A005385: Safe primes p: (p-1)/2 is also prime.
A006567: Emirps (primes whose reversal is a different prime).
A007500: Primes whose reversal in base 10 is also prime (called "palindromic primes" by D. Wells, although that name usually refers to A002385). Also called reversible primes.
A007522: Primes of the form 8n+7, that is, primes congruent to -1 mod 8.
A141112: Primes of the form 2*x^2+5*x*y-5*y^2 (as well as of the form 7*x^2+11*x*y+2*y^2).
A152079: Primes p such that A000695(p) are also prime.
A191426: Dispersion of (3+[nr]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

Number of the day: 552772

Properties of the number 552772:

552772 = 2² × 11 × 17 × 739 is the 507241^th composite number and is not squarefree.
552772 has 4 distinct prime factors, 24 divisors, 23 antidivisors and 236160 totatives.
552772 has a triangular digit sum 28 in base 10.
552772 = 138194² - 138192² = 12574² - 12552² = 8146² - 8112² = 926² - 552² is the difference of 2 nonnegative squares in 4 ways.
552772 is the difference of 2 positive pentagonal numbers in 4 ways.
552772 = 120² + 386² + 624² is the sum of 3 positive squares.
552772² = 260128² + 487740² is the sum of 2 positive squares in 1 way.
552772² is the sum of 3 positive squares.
552772 is a divisor of 373¹⁸ - 1.
552772 is palindromic in (at least) the following bases: -19, and -95.

Number of the day: 315744

Properties of the number 315744:

315744 = 2⁵ × 3 × 11 × 13 × 23 is the 288487^th composite number and is not squarefree.
315744 has 5 distinct prime factors, 96 divisors, 17 antidivisors and 84480 totatives.
315744 is the difference of 2 nonnegative squares in 32 ways.
315744 is the sum of 2 positive triangular numbers.
315744 is the difference of 2 positive pentagonal numbers in 3 ways.
315744 = 188² + 232² + 476² is the sum of 3 positive squares.
315744² = 121440² + 291456² is the sum of 2 positive squares in 1 way.
315744² is the sum of 3 positive squares.
315744 is a divisor of 967⁴ - 1.
315744 = '31574' + '4' is the concatenation of 2 semiprime numbers.
315744 is palindromic in (at least) base -99.

Number of the day: 612705

Leonhard Euler was born on this day 310 years ago.

Properties of the number 612705:

612705 = 3 × 5 × 40847 is a sphenic number and squarefree.
612705 has 3 distinct prime factors, 8 divisors, 13 antidivisors and 326768 totatives.
612705 has a semiprime digit sum 21 in base 10.
612705 has a Fibonacci digit sum 21 in base 10.
612705 has a triangular digit sum 21 in base 10.
612705 = 306353² - 306352² = 102119² - 102116² = 61273² - 61268² = 20431² - 20416² is the difference of 2 nonnegative squares in 4 ways.
612705 is the difference of 2 positive pentagonal numbers in 2 ways.
612705 = 5² + 34² + 782² is the sum of 3 positive squares.
612705² = 367623² + 490164² is the sum of 2 positive squares in 1 way.
612705² is the sum of 3 positive squares.
612705 is a divisor of 991¹⁵⁷¹ - 1.
612705 is palindromic in (at least) base 90.

The number 612705 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A264649: Number of (n+1)X(2+1) arrays of permutations of 0..n*3+2 with each element having directed index change 1,1 2,2 -1,0 or 0,-1.

Number of the day: 85106

Christiaan Huygens was born on this day 388 years ago.

Properties of the number 85106:

85106 = 2 × 7 × 6079 is a sphenic number and squarefree.
85106 has 3 distinct prime factors, 8 divisors, 5 antidivisors and 36468 totatives.
85106 has an oblong digit sum 20 in base 10.
Reversing the decimal digits of 85106 results in a sphenic number.
85106 is the sum of 2 positive triangular numbers.
85106 = 20² + 59² + 285² is the sum of 3 positive squares.
85106² is the sum of 3 positive squares.
85106 is a divisor of 1553⁶ - 1.
85106 = '85' + '106' is the concatenation of 2 semiprime numbers.
85106 is palindromic in (at least) the following bases: 88, and -65.

The number 85106 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A008488: Expansion of (1-x^6) / (1-x)^6.
A120478: Binomial(n+6,5)-binomial(n,5).
A155326: Number of ways to place zero or more nonadjacent 1,0 1,1 2,0 3,0 3,1 4,0 5,1 polyhexes in any orientation on a planar nXnXn triangular grid.

Number of the day: 917965

Properties of the number 917965:

917965 = 5 × 183593 is semiprime and squarefree.
917965 has 2 distinct prime factors, 4 divisors, 7 antidivisors and 734368 totatives.
917965 has an emirp digit sum 37 in base 10.
Reversing the decimal digits of 917965 results in an emirpimes.
917965 = 458983² - 458982² = 91799² - 91794² is the difference of 2 nonnegative squares in 2 ways.
917965 is the sum of 2 positive triangular numbers.
917965 is the difference of 2 positive pentagonal numbers in 2 ways.
917965 = 611² + 738² = 46² + 957² is the sum of 2 positive squares in 2 ways.
917965 = 75² + 242² + 924² is the sum of 3 positive squares.
917965² = 618675² + 678160² = 88044² + 913733² = 171323² + 901836² = 550779² + 734372² is the sum of 2 positive squares in 4 ways.
917965² is the sum of 3 positive squares.
917965 is a divisor of 151¹⁷³² - 1.
917965 = '9' + '17965' is the concatenation of 2 semiprime numbers.
917965 is an emirpimes in (at least) the following bases: 5, 9, 10, 11, 14, 15, 17, 18, 20, 27, 30, 31, 35, 37, 43, 47, 50, 52, 53, 62, 63, 68, 79, 80, 83, 84, 85, 86, 89, 93, 94, 96, 97, and 100.

Number of the day: 8268

Properties of the number 8268:

8268 = 2² × 3 × 13 × 53 is the 7231^th composite number and is not squarefree.
8268 has 4 distinct prime factors, 24 divisors, 11 antidivisors and 2496 totatives.
8268 = 2068² - 2066² = 692² - 686² = 172² - 146² = 92² - 14² is the difference of 2 nonnegative squares in 4 ways.
8268 is the sum of 2 positive triangular numbers.
8268 is the difference of 2 positive pentagonal numbers in 1 way.
8268 = 14² + 26² + 86² is the sum of 3 positive squares.
8268² = 4800² + 6732² = 3180² + 7632² = 1332² + 8160² = 4368² + 7020² is the sum of 2 positive squares in 4 ways.
8268² is the sum of 3 positive squares.
8268 is a divisor of 1483² - 1.
8268 is palindromic in (at least) the following bases: -22, and -57.
8268 in base 21 = iff and consists of only the digits 'f' and 'i'.
8268 in base 52 = 330 and consists of only the digits '0' and '3'.

The number 8268 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A068628: Numbers occurring twice in A068627.
A114615: Starting numbers for which the RATS sequence has eventual period 14.
A125775: Numbers n such that (5^n mod n) = (5^n mod n^2).
A127092: Numbers n such that n^2 divides 11^n-1.
A127105: Numbers n such that n^2 divides 5^n-1.
A127699: Length of period of the sequence (1^1^1^..., 2^2^2^..., 3^3^3^..., 4^4^4^..., ...) modulo n.
A184604: T(n,k) = 1/4 the number of (n+1) X (k+1) binary arrays with equal numbers of 2 X 2 subblocks with sums 1 and 3.
A226492: a(n) = n*(11*n-5)/2.
A266005: Numbers n = p_1^s_1...p_m^s_m such that (p_i^s_i - 1) | n for all 0<i<=m.
A271054: Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 259", based on the 5-celled von Neumann neighborhood.

Number of the day: 1121

Properties of the number 1121:

1121 = 19 × 59 is semiprime and squarefree.
1121 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 1044 totatives.
1121 has a prime digit sum 5 in base 10.
1121 has a Fibonacci digit sum 5 in base 10.
1121 has a prime digit product 2 in base 10.
1121 has a Fibonacci digit product 2 in base 10.
1121 has an oblong digit product 2 in base 10.
Reversing the decimal digits of 1121 results in an emirpimes.
1121 = 561² - 560² = 39² - 20² is the difference of 2 nonnegative squares in 2 ways.
1121 is the difference of 2 positive pentagonal numbers in 1 way.
1121 = 4² + 9² + 32² is the sum of 3 positive squares.
1121² is the sum of 3 positive squares.
1121 is a divisor of 1063² - 1.
1121 is an emirpimes in (at least) the following bases: 2, 3, 4, 5, 8, 9, 10, 11, 12, 14, 26, 29, 34, 37, 39, 43, 44, 47, 51, 52, 53, 54, 65, 66, 69, 70, 71, 75, 79, 80, 84, 87, 91, 94, 95, and 97.
1121 is palindromic in (at least) the following bases: 28, 32, 58, -3, -11, -35, -40, -56, -70, and -80.
1121 in base 3 = 1112112 and consists of only the digits '1' and '2'.
1121 consists of only the digits '1' and '2'.
1121 in base 23 = 22h and consists of only the digits '2' and 'h'.
1121 in base 27 = 1ee and consists of only the digits '1' and 'e'.
1121 in base 28 = 1c1 and consists of only the digits '1' and 'c'.
1121 in base 31 = 155 and consists of only the digits '1' and '5'.
1121 in base 32 = 131 and consists of only the digits '1' and '3'.

The number 1121 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: n*(3*n-2), n=0, +- 1, +- 2, +-3....
A007089: Numbers in base 3.
A007623: Integers written in factorial base.
A007651: Describe the previous term! (method B - initial term is 1).
A007931: Numbers that contain only 1's and 2's. Nonempty binary strings of length n in lexicographic order.
A015095: Carlitz-Riordan q-Catalan numbers (recurrence version) for q=10.
A028387: a(n) = n + (n+1)^2.
A045944: Rhombic matchstick numbers: n*(3*n+2).
A049345: n written in primorial base.
A052219: Numbers whose sum of digits is 5.

Number of the day: 36285

Properties of the number 36285:

36285 = 3 × 5 × 41 × 59 is the 32434^th composite number and is squarefree.
36285 has 4 distinct prime factors, 16 divisors, 21 antidivisors and 18560 totatives.
Reversing the decimal digits of 36285 results in a semiprime.
36285 = 18143² - 18142² = 6049² - 6046² = 3631² - 3626² = 1217² - 1202² = 463² - 422² = 337² - 278² = 209² - 86² = 191² - 14² is the difference of 2 nonnegative squares in 8 ways.
36285 is the difference of 2 positive pentagonal numbers in 3 ways.
36285 = 10² + 29² + 188² is the sum of 3 positive squares.
36285² = 14868² + 33099² = 7965² + 35400² = 23541² + 27612² = 21771² + 29028² is the sum of 2 positive squares in 4 ways.
36285² is the sum of 3 positive squares.
36285 is a divisor of 1063⁸ - 1.
36285 = '362' + '85' is the concatenation of 2 semiprime numbers.

The number 36285 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A082967: Numbers n such that mu(n) + mu(n+1) + mu(n+2) + mu(n+3) + mu(n+4) + mu(n+5) + mu(n+6) = 6.
A204869: Number of (n+1)X2 0..3 arrays with the permanents of all 2X2 subblocks equal and nonzero
A204874: Number of (n+1)X7 0..3 arrays with the permanents of all 2X2 subblocks equal and nonzero
A204876: T(n,k)=Number of (n+1)X(k+1) 0..3 arrays with the permanents of all 2X2 subblocks equal and nonzero
A216803: Govindarajan's triangle C arising in enumeration of multi-dimensional partitions, read by rows.

Number of the day: 508202

Élie Joseph Cartan was born on this day 148 years ago.

Properties of the number 508202:

508202 = 2 × 103 × 2467 is a sphenic number and squarefree.
508202 has 3 distinct prime factors, 8 divisors, 23 antidivisors and 251532 totatives.
508202 has an emirp digit sum 17 in base 10.
Reversing the decimal digits of 508202 results in a sphenic number.
508202 = 36² + 65² + 709² is the sum of 3 positive squares.
508202² is the sum of 3 positive squares.
508202 is a divisor of 1171³⁰⁶ - 1.
508202 is palindromic in (at least) base 90.

Number of the day: 13367

Properties of the number 13367:

13367 is the 1586^th prime.
13367 has 17 antidivisors and 13366 totatives.
13367 has an oblong digit sum 20 in base 10.
13367 has a triangular digit product 378 in base 10.
Reversing the decimal digits of 13367 results in a semiprime.
13367 = 6684² - 6683² is the difference of 2 nonnegative squares in 1 way.
13367 is the sum of 2 positive triangular numbers.
13367 is the difference of 2 positive pentagonal numbers in 1 way.
13367 is not the sum of 3 positive squares.
13367² is the sum of 3 positive squares.
13367 is a divisor of 2⁴¹ - 1.
13367 = '13' + '367' is the concatenation of 2 prime numbers.
13367 is an emirp in (at least) the following bases: 3, 4, 12, 13, 21, 28, 29, 38, 47, 53, 58, 64, 65, 69, 77, 83, and 99.
13367 is palindromic in (at least) the following bases: 9, 24, 51, 81, 82, -9, -36, -37, and -99.
13367 in base 24 = n4n and consists of only the digits '4' and 'n'.
13367 in base 36 = abb and consists of only the digits 'a' and 'b'.
13367 in base 38 = 99T and consists of only the digits '9' and 'T'.
13367 in base 50 = 5HH and consists of only the digits '5' and 'H'.
13367 in base 51 = 575 and consists of only the digits '5' and '7'.

The number 13367 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A016047: Smallest prime factor of Mersenne numbers.
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).
A055469: Primes of the form k(k+1)/2+1 (i.e. central polygonal numbers, or one more than triangular numbers).
A059236: Primes p such that x^41 = 2 has no solution mod p.
A133961: Home primes whose homeliness is greater than 3.
A133963: Home primes whose homeliness is greater than 4.
A133965: Home primes whose homeliness is greater than 5.
A133967: Home primes whose homeliness is greater than 6.
A133969: Home primes whose homeliness is greater than 7.
A133970: Home primes whose homeliness is 8.

Number of the day: 615853

Properties of the number 615853:

615853 = 7 × 97 × 907 is a sphenic number and squarefree.
615853 has 3 distinct prime factors, 8 divisors, 19 antidivisors and 521856 totatives.
615853 has a triangular digit sum 28 in base 10.
615853 = 85³ + 12³ is the sum of 2 positive cubes in 1 way.
615853 = 307927² - 307926² = 43993² - 43986² = 3223² - 3126² = 793² - 114² is the difference of 2 nonnegative squares in 4 ways.
615853 is the sum of 2 positive triangular numbers.
615853 is the difference of 2 positive pentagonal numbers in 3 ways.
615853 = 102² + 125² + 768² is the sum of 3 positive squares.
615853² = 412685² + 457128² is the sum of 2 positive squares in 1 way.
615853² is the sum of 3 positive squares.
615853 is a divisor of 523⁹⁶ - 1.
615853 = '6' + '15853' is the concatenation of 2 semiprime numbers.

Number of the day: 7467

Properties of the number 7467:

7467 = 3 × 19 × 131 is a sphenic number and squarefree.
7467 has 3 distinct prime factors, 8 divisors, 15 antidivisors and 4680 totatives.
7467 has a triangular digit product 1176 in base 10.
Reversing the decimal digits of 7467 results in a semiprime.
7467 = (18 × 19)/2 + … + (36 × 37)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
7467 = 3734² - 3733² = 1246² - 1243² = 206² - 187² = 94² - 37² is the difference of 2 nonnegative squares in 4 ways.
7467 is the difference of 2 positive pentagonal numbers in 1 way.
7467 = 1² + 35² + 79² is the sum of 3 positive squares.
7467² is the sum of 3 positive squares.
7467 is a divisor of 787⁶ - 1.
7467 = '7' + '467' is the concatenation of 2 prime numbers.
7467 is palindromic in (at least) the following bases: -23, and -41.
7467 in base 22 = f99 and consists of only the digits '9' and 'f'.
7467 in base 30 = 88r and consists of only the digits '8' and 'r'.

The number 7467 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A031583: Numbers n such that continued fraction for sqrt(n) has even period and central term 85.
A032279: Number of bracelets (turn over necklaces) of n beads of 2 colors, 5 of them black.
A043589: Numbers n such that base 3 representation has exactly 9 runs.
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.
A076425: Numbers n such that zero is never reached by iterating the mapping k -> abs(reverse(lpd(k))-reverse(gpf(k))). lpd(k) is the largest proper divisor and gpf(k) is the largest prime factor of k.
A083752: Minimal k > n such that (4k+3n)(4n+3k) is a square.
A102724: Sum of the first n pairs of consecutive primes (for example, a(3) = (2+3) + (3+5) + (5+7) = 25).
A220305: Majority value maps: number of nX3 binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal, vertical and antidiagonal neighbors in a random 0..1 nX3 array
A220308: T(n,k)=Majority value maps: number of nXk binary arrays indicating the locations of corresponding elements equal to at least half of their horizontal, vertical and antidiagonal neighbors in a random 0..1 nXk array
A272038: Somos's sequence {b(9,n)} defined in comment in A078495: a(0)=a(1)=...=a(20)=1; for n>=21, a(n)=(a(n-1)*a(n-20)+a(n-10)*a(n-11))/a(n-21).

Number of the day: 16058353

Properties of the number 16058353:

16058353 = 17 × 944609 is semiprime and squarefree.
16058353 has 2 distinct prime factors, 4 divisors, 51 antidivisors and 15113728 totatives.
16058353 has an emirp digit sum 31 in base 10.
Reversing the decimal digits of 16058353 results in an emirpimes.
16058353 = 8029177² - 8029176² = 472313² - 472296² is the difference of 2 nonnegative squares in 2 ways.
16058353 is the sum of 2 positive triangular numbers.
16058353 is the difference of 2 positive pentagonal numbers in 2 ways.
16058353 = 1928² + 3513² = 48² + 4007² is the sum of 2 positive squares in 2 ways.
16058353 = 18² + 205² + 4002² is the sum of 3 positive squares.
16058353² = 8623985² + 13546128² = 7556872² + 14169135² = 384672² + 16053745² = 7894120² + 13984047² is the sum of 2 positive squares in 4 ways.
16058353² is the sum of 3 positive squares.
16058353 is a divisor of 1721¹⁶⁸⁶⁸ - 1.
16058353 = '160583' + '53' is the concatenation of 2 prime numbers.
16058353 is an emirpimes in (at least) the following bases: 3, 6, 8, 9, 10, 11, 17, 23, 25, 26, 33, 38, 41, 43, 47, 48, 51, 62, 68, 73, 74, 77, 82, 86, 89, 93, 95, and 99.

Number of the day: 23085

Properties of the number 23085:

23085 = 3⁵ × 5 × 19 is the 20506^th composite number and is not squarefree.
23085 has 3 distinct prime factors, 24 divisors, 25 antidivisors and 11664 totatives.
23085 = 24³ + 21³ is the sum of 2 positive cubes in 1 way.
23085 is the difference of 2 nonnegative squares in 12 ways.
23085 = 5² + 34² + 148² is the sum of 3 positive squares.
23085² = 13851² + 18468² is the sum of 2 positive squares in 1 way.
23085² is the sum of 3 positive squares.
23085 is a divisor of 809⁶ - 1.
23085 is palindromic in (at least) the following bases: 2, and 8.
23085 in base 8 = 55055 and consists of only the digits '0' and '5'.

The number 23085 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005565: Number of walks on square lattice.
A035024: Expansion of 1/(1-81*x)^(1/9), related to 9-factorial numbers A045756.
A035984: Number of partitions of n into parts not of the form 21k, 21k+6 or 21k-6. Also number of partitions with at most 5 parts of size 1 and differences between parts at distance 9 are greater than 1.
A043444: Numbers n such that number of 5's in base 8 is 4.
A046320: Odd numbers divisible by exactly 7 primes (counted with multiplicity).
A085337: Numbers which are sums of two, three and four cubes.
A157948: 64n^2 - n
A158595: 361n^2 - 19.
A259380: Palindromic numbers in bases 2 and 8 written in base 10.
A276984: Sum of squares of numbers less than n that do not divide n.

Number of the day: 22750

Paul Cohen was born on this day 83 years ago.

Properties of the number 22750:

22750 = 2 × 5³ × 7 × 13 is the 20207^th composite number and is not squarefree.
22750 has 4 distinct prime factors, 32 divisors, 23 antidivisors and 7200 totatives.
22750 is the sum of 2 positive triangular numbers.
22750 is the difference of 2 positive pentagonal numbers in 9 ways.
22750 = 15² + 86² + 123² is the sum of 3 positive squares.
22750² = 8750² + 21000² = 5600² + 22050² = 11550² + 19600² = 14280² + 17710² = 2520² + 22610² = 798² + 22736² = 15582² + 16576² = 13650² + 18200² = 6370² + 21840² = 8008² + 21294² is the sum of 2 positive squares in 10 ways.
22750² is the sum of 3 positive squares.
22750 is a divisor of 307⁴ - 1.
22750 is palindromic in (at least) the following bases: 57, 94, and -57.
22750 in base 18 = 3g3g and consists of only the digits '3' and 'g'.
22750 in base 56 = 7EE and consists of only the digits '7' and 'E'.
22750 in base 57 = 707 and consists of only the digits '0' and '7'.
22750 in base 61 = 66w and consists of only the digits '6' and 'w'.

The number 22750 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A032387: Numbers n such that 75*2^n+1 is prime.
A052963: a(0)=1, a(1)=2, a(2)=5, a(n) = 3*a(n+2) - a(n+3).
A079816: Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=5, I={1}.
A095661: Fifth column (m=4) of (1,3)-Pascal triangle A095660.
A135036: Sums of the products of n consecutive pairs of numbers.
A167566: The third left hand column of triangle A167565.
A190048: Expansion of (8+6*x)/(1-x)^5
A213496: Number of (w,x,y) with all terms in {0,...,n} and x != max(|w-x|,|x-y|)
A274453: Products of distinct numbers in A052963.
A283237: a(n) = sigma_2(3*n).

Number of the day: 7824

Sophie Germain was born on this day 241 years ago.

Alexander Craig Aitken was born on this day 122 years ago.

Properties of the number 7824:

7824 = 2⁴ × 3 × 163 is the 6834^th composite number and is not squarefree.
7824 has 3 distinct prime factors, 20 divisors, 3 antidivisors and 2592 totatives.
7824 has a semiprime digit sum 21 in base 10.
7824 has a Fibonacci digit sum 21 in base 10.
7824 has a triangular digit sum 21 in base 10.
Reversing the decimal digits of 7824 results in a semiprime.
7824 = 1957² - 1955² = 980² - 976² = 655² - 649² = 493² - 485² = 332² - 320² = 175² - 151² is the difference of 2 nonnegative squares in 6 ways.
7824 is the difference of 2 positive pentagonal numbers in 1 way.
7824 = 4² + 8² + 88² is the sum of 3 positive squares.
7824² is the sum of 3 positive squares.
7824 is a divisor of 977² - 1.
7824 is palindromic in (at least) base -46.
7824 in base 5 = 222244 and consists of only the digits '2' and '4'.
7824 in base 25 = cco and consists of only the digits 'c' and 'o'.
7824 in base 39 = 55O and consists of only the digits '5' and 'O'.
7824 in base 62 = 22C and consists of only the digits '2' and 'C'.

The number 7824 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A067153: Number of regions in regular n-gon which are hexagons.
A073118: Total sum of prime parts in all partitions of n.
A087316: a(n) = Sum_{k=1..n} prime(k)^prime(n-k+1).
A128129: Expansion of (chi(-q^3)/ chi^3(-q) -1)/3 in powers of q where chi() is a Ramanujan theta function.
A132975: Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.
A132977: Expansion of q^(-1/3) * (eta(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)))^2 in powers of q.
A152760: 4 times 9-gonal numbers: a(n) = 2*n*(7*n-5).
A182977: Total number of parts that are neither the smallest part nor the largest part in all partitions of n.
A194560: G.f.: Sum_{n>=1} G_n(x)^n where G_n(x) = x + x*G_n(x)^n.
A204432: Permanent of the n-th principal submatrix of A204431.