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.
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
Offset: 0
Examples
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, ...
Links
- Andrew Howroyd, Antidiagonals n = 0..27, flattened (antidiagonals 0..12 from Alois P. Heinz)
- 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
Crossrefs
Programs
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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 *)
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