The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A257463 Number A(n,k) of factorizations of m^k into n factors, where m is a product of exactly n distinct primes and each factor is a product of k primes (counted with multiplicity); square array A(n,k), n>=0, k>=0, read by antidiagonals. 9
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 3, 10, 17, 1, 1, 1, 1, 3, 23, 93, 73, 1, 1, 1, 1, 4, 40, 465, 1417, 388, 1, 1, 1, 1, 4, 73, 1746, 19834, 32152, 2461, 1, 1, 1, 1, 5, 114, 5741, 190131, 1532489, 1016489, 18155, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS Also number of ways to partition the multiset consisting of k copies each of 1, 2, ..., n into n multisets of size k. LINKS Alois P. Heinz, Antidiagonals n = 0..12, flattened P. A. MacMahon, Combinations derived from m identical sets of n different letters and their connexion with general magic squares, Proc. London Math. Soc., 17 (1917), 25-41. The array is on page 40. Math StackExchange, Number of ways to partition 40 balls with 4 colors into 4 baskets Marko Riedel, Maple program to compute array from cycle indices EXAMPLE A(4,2) = 17: (2*3*5*7)^2 = 44100 = 15*15*14*14 = 21*15*14*10 = 21*21*10*10 = 25*14*14*9 = 25*21*14*6 = 25*21*21*4 = 35*14*10*9 = 35*15*14*6 = 35*21*10*6 = 35*21*15*4 = 35*35*6*6 = 35*35*9*4 = 49*10*10*9 = 49*15*10*6 = 49*15*15*4 = 49*25*6*6 = 49*25*9*4. A(3,3) = 10: (2*3*5)^3 = 2700 = 30*30*30 = 45*30*20 = 50*27*20 = 50*30*18 = 50*45*12 = 75*20*18 = 75*30*12 = 75*45*8 = 125*18*12 = 125*27*8. A(2,4) = 3: (2*3)^4 = 1296 = 36*36 = 54*24 = 81*16. Square array A(n,k) begins:   1, 1,   1,     1,       1,        1,         1, ...   1, 1,   1,     1,       1,        1,         1, ...   1, 1,   2,     2,       3,        3,         4, ...   1, 1,   5,    10,      23,       40,        73, ...   1, 1,  17,    93,     465,     1746,      5741, ...   1, 1,  73,  1417,   19834,   190131,   1398547, ...   1, 1, 388, 32152, 1532489, 43816115, 848597563, ... MAPLE with(numtheory): b:= proc(n, i, k) option remember; `if`(n=1, 1,       add(`if`(d>i or bigomega(d)<>k, 0,       b(n/d, d, k)), d=divisors(n)))     end: A:= (n, k)-> b(mul(ithprime(i), i=1..n)^k\$2, k): seq(seq(A(n, d-n), n=0..d), d=0..8); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n==1, 1, DivisorSum[n, If[#>i || PrimeOmega[#] != k, 0, b[n/#, #, k]]&]]; A[n_, k_] := b[p = Product[Prime[i], {i, 1, n}]^k, p, k]; Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 20 2017, translated from Maple *) CROSSREFS Columns k=0+1, 2-4 give: A000012, A002135, A254243, A268668. Rows n=0+1, 2-5 give: A000012, A008619, A257464, A253259, A253263. Cf. A257462, A257493 (ordered factorizations). Sequence in context: A265313 A106498 A093466 * A293483 A125761 A154950 Adjacent sequences:  A257460 A257461 A257462 * A257464 A257465 A257466 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Apr 24 2015 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)