Tuesday, June 27, 2017

Number of the day: 2219

Properties of the number 2219:

2219 is a cyclic number.
2219 = 7 × 317 is semiprime and squarefree.
2219 has 2 distinct prime factors, 4 divisors, 15 antidivisors and 1896 totatives.
2219 has a semiprime digit sum 14 in base 10.
2219 has a triangular digit product 36 in base 10.
Reversing the decimal digits of 2219 results in an emirpimes.
2219 = 11102 - 11092 = 1622 - 1552 is the difference of 2 nonnegative squares in 2 ways.
2219 is the difference of 2 positive pentagonal numbers in 2 ways.
2219 = 12 + 32 + 472 is the sum of 3 positive squares.
22192 = 5252 + 21562 is the sum of 2 positive squares in 1 way.
22192 is the sum of 3 positive squares.
2219 is a proper divisor of 14714 - 1.
2219 = '221' + '9' is the concatenation of 2 semiprime numbers.
2219 is an emirpimes in (at least) the following bases: 2, 4, 9, 10, 11, 12, 19, 21, 24, 28, 29, 34, 38, 40, 41, 42, 43, 45, 49, 51, 53, 62, 63, 69, 73, 76, 79, 80, 82, 83, 84, 85, 86, 93, 96, and 97.
2219 in base 23 = 44b and consists of only the digits '4' and 'b'.

The number 2219 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002212: Number of restricted hexagonal polyominoes with n cells.
A091965: Triangle read by rows: T(n,k)=number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and three types of steps H=(1,0) (left factors of 3-Motzkin steps).
A112815: Numbers n such that 7*LCM(1,2,3,...,n) equals the denominator of the n-th harmonic number H(n).
A145812: Odd positive integers a(n) such that for every odd integer m>1 there exists a unique representation of m as a sum of the form a(l)+2a(s)
A213207: Number of distinct products i*j*k over all triples (i,j,k) with |i| + |j| + |k| <= n.
A216994: Multiples of 7 such that the digit sum is divisible by 7.
A226623: Irregular array read by rows in which row n lists the smallest elements, in ascending order, of conjecturally all primitive cycles of positive integers under iteration by the Collatz-like 3x-k function, where k = A226630(n).
A226627: Irregular array read by rows. a(n) is the smallest starting value of a T_k trajectory that includes A226623(n), where T_k is the Collatz-like 3x-k function associated with A226623(n).
A237341: For k in {2,3,...,9} define a sequence as follows: a(0)=0; for n>=0, a(n+1)=a(n)+1, unless a(n) ends in k, in which case a(n+1) is obtained by replacing the last digit of a(n) with the digit(s) of k^2. This is k(4).
A259515: T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 0001 0101 0111

Monday, June 26, 2017

Number of the day: 5207010430

Properties of the number 5207010430:

5207010430 = 2 × 5 × 73 × 7132891 is the 4962794646th composite number and is squarefree.
5207010430 has 4 distinct prime factors, 16 divisors, 17 antidivisors and 2054272320 totatives.
5207010430 has a semiprime digit sum 22 in base 10.
5207010430 is the sum of 2 positive triangular numbers.
5207010430 is the difference of 2 positive pentagonal numbers in 4 ways.
5207010430 = 7142 + 11452 + 721472 is the sum of 3 positive squares.
52070104302 = 34237876802 + 39230900502 = 3851761142 + 51927446482 = 10841994322 + 50928841742 = 31242062582 + 41656083442 is the sum of 2 positive squares in 4 ways.
52070104302 is the sum of 3 positive squares.
5207010430 is a proper divisor of 9411426578 - 1.

Sunday, June 25, 2017

Number of the day: 30408

Properties of the number 30408:

30408 = 23 × 3 × 7 × 181 is the 27122th composite number and is not squarefree.
30408 has 4 distinct prime factors, 32 divisors, 11 antidivisors and 8640 totatives.
30408 has an emirpimes digit sum 15 in base 10.
30408 has a triangular digit sum 15 in base 10.
Reversing the decimal digits of 30408 results in a semiprime.
30408 = 76032 - 76012 = 38032 - 37992 = 25372 - 25312 = 12732 - 12612 = 10932 - 10792 = 5572 - 5292 = 3832 - 3412 = 2232 - 1392 is the difference of 2 nonnegative squares in 8 ways.
30408 is the sum of 2 positive triangular numbers.
30408 = 402 + 622 + 1582 is the sum of 3 positive squares.
304082 = 31922 + 302402 is the sum of 2 positive squares in 1 way.
304082 is the sum of 3 positive squares.
30408 is a proper divisor of 7434 - 1.
30408 is palindromic in (at least) the following bases: 23, -36, -51, and -64.
30408 in base 19 = 4848 and consists of only the digits '4' and '8'.
30408 in base 23 = 2bb2 and consists of only the digits '2' and 'b'.
30408 in base 35 = oss and consists of only the digits 'o' and 's'.
30408 in base 50 = C88 and consists of only the digits '8' and 'C'.

The number 30408 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A097828: Partial sums of Chebyshev sequence S(n,13)= U(n,13/2)=A078362(n).
A111078: Concerning the popular MMORPG "Runescape" by JAGeX corporation, this sequence gives the number of experience points needed for a given level in a skill.
A141724: A triangle of coefficients of a double sum skew 4th level multinomial : t(n,m,k,l)=Sum[Sum[Multinomial[n - m - k - l, m, k, l], {l, 0, k}], {k, 0, m}].
A277631: Number of aperiodic necklaces (Lyndon words) with k<=6 black beads and n-k white beads.
A288564: Number of connected one-sided arrangements of n pseudo-circles in the affine plane, in the sense that the union of the solid pseudo-circles is a connected set.

Saturday, June 24, 2017

Number of the day: 7170

Properties of the number 7170:

7170 = 2 × 3 × 5 × 239 is the 6253th composite number and is squarefree.
7170 has 4 distinct prime factors, 16 divisors, 9 antidivisors and 1904 totatives.
7170 has an emirpimes digit sum 15 in base 10.
7170 has a triangular digit sum 15 in base 10.
7170 is the difference of 2 positive pentagonal numbers in 2 ways.
7170 = 52 + 162 + 832 is the sum of 3 positive squares.
71702 = 43022 + 57362 is the sum of 2 positive squares in 1 way.
71702 is the sum of 3 positive squares.
7170 is a proper divisor of 4792 - 1.
7170 is palindromic in (at least) the following bases: 14, 56, 67, -36, and -64.
7170 in base 7 = 26622 and consists of only the digits '2' and '6'.
7170 in base 14 = 2882 and consists of only the digits '2' and '8'.
7170 in base 25 = bbk and consists of only the digits 'b' and 'k'.
7170 in base 34 = 66u and consists of only the digits '6' and 'u'.
7170 in base 55 = 2KK and consists of only the digits '2' and 'K'.
7170 in base 56 = 2G2 and consists of only the digits '2' and 'G'.

The number 7170 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A054341: Row sums of triangle A054336 (central binomial convolutions).
A055574: n satisfying sigma(n+1) = sigma(n-1).
A067347: Square array read by antidiagonals: T(n,k)=(T(n,k-1)*n^2-Catalan(k-1)*n)/(n-1) with a(n,0)=1 and a(1,k)=Catalan(k) where Catalan(k)=C(2k,k)/(k+1)=A000108(k).
A076036: G.f.: 1/(1 - 5*x*C(x)), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) = g.f. for the Catalan numbers A000108.
A125205: Triangular array T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).
A125206: Triangular array T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs obtained from the complete labeled graph K_n by removing k edges.
A139274: a(n) = n*(8*n-1).
A239832: Number of partitions of n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even.
A240192: T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or three plus the sum of the elements diagonally to its northwest, modulo 4
A269467: T(n,k)=Number of length-n 0..k arrays with no repeated value equal to the previous repeated value.

Thursday, June 22, 2017

Number of the day: 1509

Hermann Minkowski was born on this day 153 years ago.

Properties of the number 1509:

1509 is a cyclic number.
1509 = 3 × 503 is semiprime and squarefree.
1509 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 1004 totatives.
1509 has an emirpimes digit sum 15 in base 10.
1509 has a triangular digit sum 15 in base 10.
1509 has sum of divisors equal to 2016 which is a triangular number.
Reversing the decimal digits of 1509 results in a sphenic number.
1509 = 7552 - 7542 = 2532 - 2502 is the difference of 2 nonnegative squares in 2 ways.
1509 is the sum of 2 positive triangular numbers.
1509 is the difference of 2 positive pentagonal numbers in 1 way.
1509 = 82 + 172 + 342 is the sum of 3 positive squares.
15092 is the sum of 3 positive squares.
1509 is a proper divisor of 7251 - 1.
1509 is an emirpimes in (at least) the following bases: 4, 6, 9, 11, 12, 15, 18, 19, 20, 22, 24, 32, 36, 38, 41, 42, 47, 49, 55, 57, 59, 60, 64, 65, 71, 75, 77, 83, 84, 87, 91, 93, 96, 98, and 99.
1509 is palindromic in (at least) the following bases: 16, 29, -52, and -58.
1509 in base 16 = 5e5 and consists of only the digits '5' and 'e'.
1509 in base 28 = 1pp and consists of only the digits '1' and 'p'.
1509 in base 29 = 1n1 and consists of only the digits '1' and 'n'.
1509 in base 38 = 11R and consists of only the digits '1' and 'R'.

The number 1509 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A005228: Sequence and first differences (A030124) together list all positive numbers exactly once.
A006832: Discriminants of totally real cubic fields.
A026105: Triangle T read by rows: differences of Motzkin triangle (A026300).
A059993: Pinwheel numbers: a(n) = 2*n^2 + 6*n + 1.
A094612: Fundamental discriminants of real quadratic number fields with class number 3.
A165652: Number of disconnected 2-regular graphs on n vertices.
A202124: T(n,k)=Number of -k..k arrays of n elements with first, second and third differences also in -k..k
A224146: T(n,k)=Number of nXk 0..1 arrays with rows and antidiagonals unimodal and columns nondecreasing
A241306: T(n,k)=Number of nXk 0..3 arrays with no element equal to zero plus the sum of elements to its left or one plus the sum of the elements above it or zero plus the sum of the elements diagonally to its northwest or zero plus the sum of the elements antidiagonally to its northeast, modulo 4
A279567: Number of length n inversion sequences avoiding the patterns 100, 110, 120, and 210.

Wednesday, June 21, 2017

Number of the day: 9348

Properties of the number 9348:

9348 = 22 × 3 × 19 × 41 is the 8190th composite number and is not squarefree.
9348 has 4 distinct prime factors, 24 divisors, 11 antidivisors and 2880 totatives.
Reversing the decimal digits of 9348 results in a sphenic number.
9348 = 23382 - 23362 = 7822 - 7762 = 1422 - 1042 = 982 - 162 is the difference of 2 nonnegative squares in 4 ways.
9348 is the sum of 2 positive triangular numbers.
9348 is the difference of 2 positive pentagonal numbers in 1 way.
9348 = 22 + 402 + 882 is the sum of 3 positive squares.
93482 = 20522 + 91202 is the sum of 2 positive squares in 1 way.
93482 is the sum of 3 positive squares.
9348 is a proper divisor of 15592 - 1.
9348 is palindromic in (at least) the following bases: -13, -22, -26, and -33.
9348 in base 25 = enn and consists of only the digits 'e' and 'n'.
9348 in base 32 = 944 and consists of only the digits '4' and '9'.
9348 in base 36 = 77o and consists of only the digits '7' and 'o'.

The number 9348 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001610: a(n) = a(n-1) + a(n-2) + 1.
A031367: Inflation orbit counts.
A099925: a(n) = Lucas(n) + (-1)^n.
A120019: Square table, read by antidiagonals, of self-compositions of A120010.
A120020: Coefficients of x^n in the n-th iteration of the g.f. of A120010: a(n) = [x^n] { (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) } for n>=1.
A179249: Numbers n that have 9 terms in their Zeckendorf representation.
A197218: Phi(Lucas(n)).
A209796: T(n,k)=Half the number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having exactly one duplicate clockwise edge difference
A214980: Positions of zeros in A214979.
A231089: Initial members of abundant quadruplets, i.e., values of n such that (n, n+2, n+4, n+6) are all abundant numbers.

Tuesday, June 20, 2017

Number of the day: 746010

Properties of the number 746010:

746010 = 2 × 35 × 5 × 307 is the 686066th composite number and is not squarefree.
746010 has 4 distinct prime factors, 48 divisors, 25 antidivisors and 198288 totatives.
746010 = 203 + 433 + 873 = 423 + 633 + 753 is the sum of 3 positive cubes in 2 ways.
746010 is the difference of 2 positive pentagonal numbers in 1 way.
746010 = 552 + 1242 + 8532 is the sum of 3 positive squares.
7460102 = 4476062 + 5968082 is the sum of 2 positive squares in 1 way.
7460102 is the sum of 3 positive squares.
746010 is a proper divisor of 63127 - 1.
746010 is palindromic in (at least) base -97.

Monday, June 19, 2017

Number of the day: 641

Blaise Pascal was born on this day 394 years ago.

Properties of the number 641:

641 is a cyclic number.
641 and 643 form a twin prime pair.
641 has 7 antidivisors and 640 totatives.
641 has a prime digit sum 11 in base 10.
641 has sum of divisors equal to 642 which is a sphenic number.
Reversing the decimal digits of 641 results in a semiprime.
641 = 3212 - 3202 is the difference of 2 nonnegative squares in 1 way.
641 is the difference of 2 positive pentagonal numbers in 1 way.
641 = 42 + 252 is the sum of 2 positive squares in 1 way.
641 = 62 + 112 + 222 is the sum of 3 positive squares.
6412 = 2002 + 6092 is the sum of 2 positive squares in 1 way.
6412 is the sum of 3 positive squares.
641 is a proper divisor of 4874 - 1.
641 is an emirp in (at least) the following bases: 3, 9, 11, 13, 15, 22, 27, 29, 36, 37, 45, 47, 48, 53, 54, 57, 59, 61, 71, 75, 77, 79, 82, 83, 85, 86, 87, 88, 89, and 90.
641 is palindromic in (at least) the following bases: 20, -12, -13, -32, -40, -64, and -80.
641 in base 11 = 533 and consists of only the digits '3' and '5'.
641 in base 12 = 455 and consists of only the digits '4' and '5'.
641 in base 14 = 33b and consists of only the digits '3' and 'b'.
641 in base 19 = 1ee and consists of only the digits '1' and 'e'.
641 in base 20 = 1c1 and consists of only the digits '1' and 'c'.

The number 641 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A001097: Twin primes.
A001359: Lesser of twin primes.
A005384: Sophie Germain primes p: 2p+1 is also prime.
A005846: Primes of the form n^2 + n + 41.
A007519: Primes of form 8n+1, that is, primes congruent to 1 mod 8.
A023201: Sexy primes: numbers n such that n and n + 6 are both prime.
A104272: Ramanujan primes R_n: a(n) is the smallest number such that if x >= a(n), then pi(x) - pi(x/2) >= n, where pi(x) is the number of primes <= x.
A106856: Primes of the form x^2+xy+2y^2, with x and y nonnegative.
A212959: Number of (w,x,y) such that w,x,y are all in {0,...,n} and |w-x| = |x-y|.
A235266: Primes whose base 2 representation is also the base 3 representation of a prime.

Saturday, June 17, 2017

Number of the day: 53982

Properties of the number 53982:

53982 = 2 × 32 × 2999 is the 48483th composite number and is not squarefree.
53982 has 3 distinct prime factors, 12 divisors, 21 antidivisors and 17988 totatives.
53982 is the difference of 2 positive pentagonal numbers in 2 ways.
53982 = 382 + 532 + 2232 is the sum of 3 positive squares.
539822 is the sum of 3 positive squares.
53982 is a proper divisor of 731499 - 1.
53982 is palindromic in (at least) the following bases: 85, -45, and -70.
53982 in base 44 = Rcc and consists of only the digits 'R' and 'c'.

The number 53982 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A249058: a(n) = number of primes less than the square root of the (2^n)-th prime

Friday, June 16, 2017

Number of the day: 580667

Properties of the number 580667:

580667 is a cyclic number.
580667 = 29 × 20023 is semiprime and squarefree.
580667 has 2 distinct prime factors, 4 divisors, 13 antidivisors and 560616 totatives.
580667 = 2903342 - 2903332 = 100262 - 99972 is the difference of 2 nonnegative squares in 2 ways.
580667 is the difference of 2 positive pentagonal numbers in 2 ways.
580667 = 52 + 392 + 7612 is the sum of 3 positive squares.
5806672 = 4004602 + 4204832 is the sum of 2 positive squares in 1 way.
5806672 is the sum of 3 positive squares.
580667 is a proper divisor of 563564 - 1.
580667 is an emirpimes in (at least) the following bases: 2, 3, 5, 6, 9, 11, 13, 18, 19, 23, 24, 27, 29, 32, 38, 42, 48, 49, 53, 60, 67, 68, 71, 72, 73, 74, 76, 79, 84, 87, 89, 90, 94, and 98.
580667 is palindromic in (at least) base -93.

The number 580667 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequence (among others):

Sequence number and description below are taken from OEIS.
A244246: Number of partitions of n into 10 parts such that every i-th smallest part (counted with multiplicity) is different from i.

Thursday, June 15, 2017

Number of the day: 92756230379

Properties of the number 92756230379:

92756230379 is a cyclic number.
92756230379 = 1231 × 75350309 is semiprime and squarefree.
92756230379 has 2 distinct prime factors, 4 divisors, 33 antidivisors and 92680878840 totatives.
92756230379 has a prime digit sum 53 in base 10.
Reversing the decimal digits of 92756230379 results in a sphenic number.
92756230379 = 463781151902 - 463781151892 = 376757702 - 376745392 is the difference of 2 nonnegative squares in 2 ways.
92756230379 is the difference of 2 positive pentagonal numbers in 2 ways.
92756230379 = 9732 + 19932 + 3045512 is the sum of 3 positive squares.
927562303792 = 400560752602 + 836613955712 is the sum of 2 positive squares in 1 way.
927562303792 is the sum of 3 positive squares.
92756230379 is a proper divisor of 12720550084 - 1.
92756230379 = '9' + '2756230379' is the concatenation of 2 semiprime numbers.
92756230379 is an emirpimes in (at least) the following bases: 2, 5, 6, 9, 11, 13, 17, 20, 22, 23, 26, 36, 43, 48, 50, 54, 60, 61, 62, 64, 69, 71, 74, 77, 81, 88, 95, 96, and 100.

Wednesday, June 14, 2017

Number of the day: 16634598676

Andrey Andreyevich Markov was born on this day 161 years ago.

Properties of the number 16634598676:

16634598676 = 22 × 37 × 43 × 97 × 26947 is composite and not squarefree.
16634598676 has 5 distinct prime factors, 48 divisors, 79 antidivisors and 7822531584 totatives.
16634598676 has a prime digit sum 61 in base 10.
Reversing the decimal digits of 16634598676 results in a prime.
16634598676 = (14919 × 14920)/2 + … + (15066 × 15067)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
16634598676 = 41586496702 - 41586496682 = 1123959742 - 1123959002 = 967128262 - 967127402 = 428727742 - 428725802 = 26154502 - 26122682 = 11623102 - 11551322 = 10012102 - 9928682 = 1812742 - 1273802 is the difference of 2 nonnegative squares in 8 ways.
16634598676 is the difference of 2 positive pentagonal numbers in 8 ways.
16634598676 = 6262 + 23642 + 1289522 is the sum of 3 positive squares.
166345986762 = 111468960202 + 123473309762 = 80646981602 + 145489008762 = 65398213242 + 152951172002 = 53950049762 + 157354311802 is the sum of 2 positive squares in 4 ways.
166345986762 is the sum of 3 positive squares.
16634598676 is a proper divisor of 1543504 - 1.
16634598676 = '1663459867' + '6' is the concatenation of 2 semiprime numbers.

Tuesday, June 13, 2017

Number of the day: 172614

John Forbes Nash, Jr. was born on this day 89 years ago.

Properties of the number 172614:

172614 = 2 × 3 × 13 × 2213 is the 156892th composite number and is squarefree.
172614 has 4 distinct prime factors, 16 divisors, 7 antidivisors and 53088 totatives.
172614 has a semiprime digit sum 21 in base 10.
172614 has a Fibonacci digit sum 21 in base 10.
172614 has a triangular digit sum 21 in base 10.
172614 is the sum of 2 positive triangular numbers.
172614 is the difference of 2 positive pentagonal numbers in 4 ways.
172614 = 102 + 172 + 4152 is the sum of 3 positive squares.
1726142 = 796862 + 1531202 = 663902 + 1593362 = 526142 + 1644002 = 146642 + 1719902 is the sum of 2 positive squares in 4 ways.
1726142 is the sum of 3 positive squares.
172614 is a proper divisor of 109314 - 1.
172614 = '1726' + '14' is the concatenation of 2 semiprime numbers.

Monday, June 12, 2017

Number of the day: 73084

Vladimir Igorevich Arnold was born on this day 80 years ago.

Properties of the number 73084:

73084 = 22 × 112 × 151 is the 65856th composite number and is not squarefree.
73084 has 3 distinct prime factors, 18 divisors, 25 antidivisors and 33000 totatives.
73084 has a semiprime digit sum 22 in base 10.
73084 = 182722 - 182702 = 16722 - 16502 = 2722 - 302 is the difference of 2 nonnegative squares in 3 ways.
73084 is the sum of 2 positive triangular numbers.
73084 is the difference of 2 positive pentagonal numbers in 4 ways.
73084 is not the sum of 3 positive squares.
730842 is the sum of 3 positive squares.
73084 is a proper divisor of 16936 - 1.
73084 is palindromic in (at least) base -12.
73084 in base 45 = a44 and consists of only the digits '4' and 'a'.

Sunday, June 11, 2017

Number of the day: 21429

Properties of the number 21429:

21429 = 32 × 2381 is the 19023th composite number and is not squarefree.
21429 has 2 distinct prime factors, 6 divisors, 7 antidivisors and 14280 totatives.
21429 has a Fibonacci digit product 144 in base 10.
21429 = 13 + 213 + 233 is the sum of 3 positive cubes in 1 way.
21429 = 107152 - 107142 = 35732 - 35702 = 11952 - 11862 is the difference of 2 nonnegative squares in 3 ways.
21429 is the sum of 2 positive triangular numbers.
21429 is the difference of 2 positive pentagonal numbers in 1 way.
21429 = 1022 + 1052 is the sum of 2 positive squares in 1 way.
21429 = 72 + 82 + 1462 is the sum of 3 positive squares.
214292 = 6212 + 214202 is the sum of 2 positive squares in 1 way.
214292 is the sum of 3 positive squares.
21429 is a proper divisor of 102121 - 1.
21429 = '2' + '1429' is the concatenation of 2 prime numbers.
21429 is palindromic in (at least) the following bases: -32, and -51.
21429 in base 11 = 15111 and consists of only the digits '1' and '5'.
21429 in base 28 = r99 and consists of only the digits '9' and 'r'.
21429 in base 34 = ii9 and consists of only the digits '9' and 'i'.
21429 in base 35 = hh9 and consists of only the digits '9' and 'h'.

The number 21429 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A031846: Period of continued fraction for sqrt(n) contains exactly 78 ones.
A037101: Trajectory of 3 under map n->7n+1 if n odd, n->n/2 if n even
A271412: Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 358", based on the 5-celled von Neumann neighborhood.
A271623: a(0)=7; a(n) = 7*a(n-1) + 1 if a(n-1) is odd, a(n) = a(n-1)/2 otherwise.

Saturday, June 10, 2017

Number of the day: 2716

Properties of the number 2716:

2716 = 22 × 7 × 97 is the 2319th composite number and is not squarefree.
2716 has 3 distinct prime factors, 12 divisors, 5 antidivisors and 1152 totatives.
2716 = 173 - 133 is the difference of 2 positive cubes in 1 way.
2716 = (15 × 16)/2 + … + (26 × 27)/2 is the sum of at least 2 consecutive triangular numbers in 1 way.
2716 = 6802 - 6782 = 1042 - 902 is the difference of 2 nonnegative squares in 2 ways.
2716 is the sum of 2 positive triangular numbers.
2716 is the difference of 2 positive pentagonal numbers in 2 ways.
2716 is not the sum of 3 positive squares.
27162 = 18202 + 20162 is the sum of 2 positive squares in 1 way.
27162 is the sum of 3 positive squares.
2716 is a proper divisor of 11632 - 1.
2716 is palindromic in (at least) the following bases: 24, 96, -21, and -46.
2716 in base 15 = c11 and consists of only the digits '1' and 'c'.
2716 in base 24 = 4h4 and consists of only the digits '4' and 'h'.

The number 2716 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002895: Domb numbers: number of 2n-step polygons on diamond lattice.
A007655: Standard deviation of A007654.
A063490: (2*n-1)*(7*n^2-7*n+6)/6.
A075232: Numbers n such that n^9 is an interprime = average of two successive primes.
A076454: Sum of numbers that can be written as t*n + u*(n+1) for nonnegative integers t,u in exactly one way.
A144138: Chebyshev polynomial of the second kind U(3,n).
A180281: Triangle read by rows: T(n,k) = number of arrangements of n indistinguishable balls in n boxes with the maximum number of balls in any box equal to k.
A181061: a(n) is the smallest positive number such that the decimal digits of n*a(n) are all 0, 1 or 2.
A196636: T(n,k)=Number of nXk 0..4 arrays with each element x equal to the number of its horizontal and vertical neighbors equal to 1,3,2,0,4 for x=0,1,2,3,4
A245397: A(n,k) is the sum of k-th powers of coefficients in full expansion of (z_1+z_2+...+z_n)^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Friday, June 9, 2017

Number of the day: 1906

John Edensor Littlewood was born on this day 132 years ago.

Properties of the number 1906:

1906 = 2 × 953 is semiprime and squarefree.
1906 has 2 distinct prime factors, 4 divisors, 9 antidivisors and 952 totatives.
1906 has sum of divisors equal to 2862 which is an oblong number.
Reversing the decimal digits of 1906 results in a prime.
1906 is the sum of 2 positive triangular numbers.
1906 is the difference of 2 positive pentagonal numbers in 2 ways.
1906 = 152 + 412 is the sum of 2 positive squares in 1 way.
1906 = 132 + 212 + 362 is the sum of 3 positive squares.
19062 = 12302 + 14562 is the sum of 2 positive squares in 1 way.
19062 is the sum of 3 positive squares.
1906 is a proper divisor of 4317 - 1.
1906 = '190' + '6' is the concatenation of 2 triangular numbers.
1906 is an emirpimes in (at least) the following bases: 3, 5, 7, 11, 13, 18, 21, 23, 25, 29, 31, 32, 33, 35, 38, 46, 47, 48, 49, 51, 54, 56, 59, 67, 70, 71, 72, 74, 76, 81, 83, 84, 86, 89, 90, 91, 96, and 98.
1906 is palindromic in (at least) the following bases: 28, -19, and -34.
1906 in base 3 = 2121121 and consists of only the digits '1' and '2'.
1906 in base 16 = 772 and consists of only the digits '2' and '7'.
1906 in base 19 = 556 and consists of only the digits '5' and '6'.
1906 in base 27 = 2gg and consists of only the digits '2' and 'g'.
1906 in base 28 = 2c2 and consists of only the digits '2' and 'c'.
1906 in base 43 = 11E and consists of only the digits '1' and 'E'.

The number 1906 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A006315: Numbers n such that n^32 + 1 is prime.
A077065: Semiprimes of form prime - 1.
A090495: Numbers n such that numerator(Bernoulli(2*n)/(2*n)) is different from numerator(Bernoulli(2*n)/(2*n*(2*n-1))).
A098185: If f(x) = (sum of unitary proper divisors of x) = A063919(x) is iterated, the iteration may lead to a fixed point which is either equals 0 or it is from A002827, a unitary perfect number > 1: 6,60,90,87360... Here initial values are collected for which the iteration ends in a unitary perfect number > 1.
A120186: a(n)=ceiling( sum_{i=1..n-1} a(i)/7), a(1)=1.
A206128: T(n,k)=Number of (n+1)X(k+1) 0..2 arrays with every 2X2 subblock having nonzero determinant and having the same number of clockwise edge increases as its horizontal and vertical neighbors
A217135: Numbers n such that 3^n - 8 is prime.
A229717: T(n,k)=Number of arrays of length n that are sums of k consecutive elements of length n+k-1 permutations of 0..n+k-2, and no two consecutive rises or falls in the latter permutation
A238340: Number of partitions of 4n into 4 parts.
A281205: T(n,k)=Number of nXk 0..1 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

Thursday, June 8, 2017

Number of the day: 21414

Properties of the number 21414:

21414 = 2 × 3 × 43 × 83 is the 19009th composite number and is squarefree.
21414 has 4 distinct prime factors, 16 divisors, 9 antidivisors and 6888 totatives.
21414 has an oblong digit sum 12 in base 10.
Reversing the decimal digits of 21414 results in an oblong number.
21414 is the difference of 2 positive pentagonal numbers in 2 ways.
21414 = 102 + 172 + 1452 is the sum of 3 positive squares.
214142 is the sum of 3 positive squares.
21414 is a proper divisor of 13276 - 1.
21414 = '214' + '14' is the concatenation of 2 semiprime numbers.
21414 is palindromic in (at least) the following bases: 41, -30, and -40.
21414 in base 9 = 32333 and consists of only the digits '2' and '3'.
21414 in base 13 = 9993 and consists of only the digits '3' and '9'.
21414 in base 30 = nno and consists of only the digits 'n' and 'o'.
21414 in base 39 = E33 and consists of only the digits '3' and 'E'.
21414 in base 41 = CUC and consists of only the digits 'C' and 'U'.

The number 21414 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A032240: Number of identity bracelets of n beads of 3 colors.
A043468: Numbers n such that number of 3's in base 9 is 4.
A074303: Sum of squares of digits of n is equal to the largest prime factor of n reversed, where the largest prime factor is not a palindrome.
A167690: The even composites n such that n=q*g*j*y and q+g=j*y where q,g,j,y are primes.
A200075: G.f. satisfies: A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)^3).
A211850: Number of nonnegative integer arrays of length 2n+5 with new values 0 upwards introduced in order, no three adjacent elements all unequal, and containing the value n+1
A260761: Number of (n+2)X(2+2) 0..1 arrays with each 3X3 subblock having clockwise perimeter pattern 00000001 00000011 or 00010011
A283003: Intersection of A003052 and A283002.

Wednesday, June 7, 2017

Number of the day: 193526105

Properties of the number 193526105:

193526105 = 5 × 181 × 281 × 761 is the 182785984th composite number and is squarefree.
193526105 has 4 distinct prime factors, 16 divisors, 67 antidivisors and 153216000 totatives.
Reversing the decimal digits of 193526105 results in a sphenic number.
193526105 = 967630532 - 967630522 = 193526132 - 193526082 = 5346932 - 5345122 = 3444932 - 3442122 = 1275332 - 1267722 = 1073732 - 1064682 = 695732 - 681682 = 273332 - 235282 is the difference of 2 nonnegative squares in 8 ways.
193526105 is the difference of 2 positive pentagonal numbers in 8 ways.
193526105 = 32932 + 135162 = 18562 + 137872 = 97572 + 99162 = 88372 + 107442 = 25962 + 136672 = 11472 + 138642 = 93762 + 102772 = 92362 + 104032 is the sum of 2 positive squares in 8 ways.
193526105 = 1352 + 3622 + 139062 is the sum of 3 positive squares.
1935261052 = 1101928002 + 1590908552 = 928838552 + 1697791002 = 1262841452 + 1466447002 = 130404962 + 1930862472 = 73002732 + 1933883642 = 275603762 + 1915535932 = 611570042 + 1836087532 = 415452732 + 1890141362 = 800929672 + 1761745442 = 1018948552 + 1645290002 = 840609002 + 1743161452 = 1186029002 + 1529238552 = 172015162 + 1927601132 = 31280072 + 1935008242 = 373409672 + 1898894562 = 704862872 + 1802332842 = 511773442 + 1866366332 = 890163762 + 1718384072 = 1182011452 + 1532346002 = 1014627002 + 1647958552 = 1336335002 + 1399801452 = 177073532 + 1927143042 = 26201562 + 1935083672 = 229187132 + 1921642162 = 516669932 + 1865016762 = 318040162 + 1908948872 = 709590642 + 1800476732 = 1080287642 + 1605681772 = 905766972 + 1710210962 = 1242871432 + 1483410242 = 104251002 + 1932451052 = 99178952 + 1932718002 = 301513002 + 1911628952 = 1238973962 + 1486667032 = 1076069832 + 1608511442 = 1348395362 + 1388187772 = 1161156632 + 1548208842 = 992222242 + 1661544572 = 1317260562 + 1417765832 = 203148952 + 1924569002 is the sum of 2 positive squares in 40 ways.
1935261052 is the sum of 3 positive squares.
193526105 is a proper divisor of 67120 - 1.

Tuesday, June 6, 2017

Number of the day: 50633

Aleksandr Mikhailovich Lyapunov was born on this day 160 years ago.

Properties of the number 50633:

50633 is a cyclic number.
50633 = 11 × 4603 is semiprime and squarefree.
50633 has 2 distinct prime factors, 4 divisors, 17 antidivisors and 46020 totatives.
50633 has an emirp digit sum 17 in base 10.
50633 = 253172 - 253162 = 23072 - 22962 is the difference of 2 nonnegative squares in 2 ways.
50633 is the difference of 2 positive pentagonal numbers in 2 ways.
50633 = 42 + 212 + 2242 is the sum of 3 positive squares.
506332 is the sum of 3 positive squares.
50633 is a proper divisor of 17915 - 1.
50633 is an emirpimes in (at least) the following bases: 2, 3, 4, 9, 14, 19, 21, 26, 30, 32, 33, 35, 36, 37, 41, 44, 46, 49, 51, 55, 58, 62, 65, 67, 68, 69, 73, 77, 82, 86, 89, 91, 94, 97, and 98.
50633 is palindromic in (at least) the following bases: 53, 66, 71, and -81.
50633 in base 37 = aaH and consists of only the digits 'H' and 'a'.
50633 in base 39 = XBB and consists of only the digits 'B' and 'X'.
50633 in base 52 = Ibb and consists of only the digits 'I' and 'b'.
50633 in base 53 = I1I and consists of only the digits '1' and 'I'.

Monday, June 5, 2017

Number of the day: 725066

Properties of the number 725066:

725066 = 2 × 43 × 8431 is a sphenic number and squarefree.
725066 has 3 distinct prime factors, 8 divisors, 7 antidivisors and 354060 totatives.
725066 has an emirpimes digit sum 26 in base 10.
Reversing the decimal digits of 725066 results in a sphenic number.
725066 = 442 + 1712 + 8332 is the sum of 3 positive squares.
7250662 is the sum of 3 positive squares.
725066 is a proper divisor of 937210 - 1.

Sunday, June 4, 2017

Number of the day: 54914

Properties of the number 54914:

54914 = 2 × 27457 is semiprime and squarefree.
54914 has 2 distinct prime factors, 4 divisors, 5 antidivisors and 27456 totatives.
54914 has a prime digit sum 23 in base 10.
54914 has sum of divisors equal to 82374 which is a sphenic number.
Reversing the decimal digits of 54914 results in an emirpimes.
54914 = 252 + 2332 is the sum of 2 positive squares in 1 way.
54914 = 232 + 322 + 2312 is the sum of 3 positive squares.
549142 = 116502 + 536642 is the sum of 2 positive squares in 1 way.
549142 is the sum of 3 positive squares.
54914 is a proper divisor of 10311 - 1.
54914 is an emirpimes in (at least) the following bases: 4, 6, 8, 9, 10, 13, 15, 16, 17, 19, 20, 22, 24, 25, 26, 29, 31, 34, 35, 37, 41, 43, 50, 55, 56, 63, 64, 69, 70, 76, 77, 78, 79, 81, 84, 86, 88, 91, 92, 95, 98, and 99.
54914 is palindromic in (at least) the following bases: 40, 61, -13, and -79.
54914 in base 40 = YCY and consists of only the digits 'C' and 'Y'.
54914 in base 60 = FFE and consists of only the digits 'E' and 'F'.
54914 in base 61 = EkE and consists of only the digits 'E' and 'k'.

The number 54914 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A188849: Number of nX7 binary arrays without the pattern 0 0 1 vertically, antidiagonally or horizontally
A188851: T(n,k)=Number of nXk binary arrays without the pattern 0 0 1 vertically, antidiagonally or horizontally
A188852: Number of 3Xn binary arrays without the pattern 0 0 1 vertically, antidiagonally or horizontally

Saturday, June 3, 2017

Number of the day: 8660

Properties of the number 8660:

8660 = 22 × 5 × 433 is the 7582th composite number and is not squarefree.
8660 has 3 distinct prime factors, 12 divisors, 9 antidivisors and 3456 totatives.
8660 has an oblong digit sum 20 in base 10.
8660 = 21662 - 21642 = 4382 - 4282 is the difference of 2 nonnegative squares in 2 ways.
8660 is the difference of 2 positive pentagonal numbers in 2 ways.
8660 = 442 + 822 = 142 + 922 is the sum of 2 positive squares in 2 ways.
8660 = 42 + 302 + 882 is the sum of 3 positive squares.
86602 = 29002 + 81602 = 25762 + 82682 = 47882 + 72162 = 51962 + 69282 is the sum of 2 positive squares in 4 ways.
86602 is the sum of 3 positive squares.
8660 is a proper divisor of 1794 - 1.
8660 is palindromic in (at least) the following bases: -74, and -78.
8660 in base 46 = 44C and consists of only the digits '4' and 'C'.

The number 8660 belongs to the following On-Line Encyclopedia of Integer Sequences (OEIS) sequences (among others):

Sequence numbers and descriptions below are taken from OEIS.
A002627: a(n) = n*a(n-1) + 1, a(0) = 0.
A003430: Number of unlabeled N-free posets (i.e. generated by unions and sums) with n nodes.
A059922: Each term in the table is the product of the two terms above it + 1.
A061658: In base 4 n and n^2 contain the same digits in the same proportion.
A087610: Number of (-1,0,1) polynomials of degree-n irreducible over the integers.
A180223: a(n) = (11*n^2 - 7*n)/2.
A207061: Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+433)^2 = y^2.
A219040: Numbers n such that 3^n + 20 is prime.
A231246: T(n,k)=Number of nXk 0..3 arrays x(i,j) with each element horizontally, diagonally or antidiagonally next to at least one element with value (x(i,j)+1) mod 4, no adjacent elements equal, and upper left element zero
A260294: T(n,k)=Number of (n+2)X(k+2) 0..1 arrays with each 3X3 subblock having clockwise perimeter pattern 00000001 00000011 or 00001001

Friday, June 2, 2017

Number of the day: 586598

Properties of the number 586598:

586598 = 2 × 37 × 7927 is a sphenic number and squarefree.
586598 has 3 distinct prime factors, 8 divisors, 39 antidivisors and 285336 totatives.
586598 has a prime digit sum 41 in base 10.
586598 = 22 + 372 + 7652 is the sum of 3 positive squares.
5865982 = 1902482 + 5548902 is the sum of 2 positive squares in 1 way.
5865982 is the sum of 3 positive squares.
586598 is a proper divisor of 5931321 - 1.
586598 = '58' + '6598' is the concatenation of 2 semiprime numbers.

Thursday, June 1, 2017

Number of the day: 15275180

Properties of the number 15275180:

15275180 = 22 × 5 × 17 × 44927 is the 14287822th composite number and is not squarefree.
15275180 has 4 distinct prime factors, 24 divisors, 15 antidivisors and 5750528 totatives.
15275180 has a prime digit sum 29 in base 10.
15275180 = 38187962 - 38187942 = 7637642 - 7637542 = 2246522 - 2246182 = 450122 - 448422 is the difference of 2 nonnegative squares in 4 ways.
15275180 is the difference of 2 positive pentagonal numbers in 4 ways.
15275180 = 382 + 1302 + 39062 is the sum of 3 positive squares.
152751802 = 71883202 + 134781002 = 23362042 + 150954722 = 64694882 + 138375162 = 91651082 + 122201442 is the sum of 2 positive squares in 4 ways.
152751802 is the sum of 3 positive squares.
15275180 is a proper divisor of 8912836 - 1.