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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.
12
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
0,13
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
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.
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.
Main diagonal gives A334286.
Cf. A257462, A257493 (ordered factorizations).
Sequence in context: A265313 A106498 A093466 * A293483 A125761 A154950
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 24 2015
STATUS
approved