A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.

1, 1, 1, 3, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 2, 9, 1, 4, 1, 4, 2, 2, 1, 7, 3, 2, 4, 4, 1, 5, 1, 8, 2, 2, 2, 12, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 3, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 22, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 9, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 12, 1, 5, 1, 7, 5

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..10080

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Index entries for sequences computed from exponents in factorization of n

Formula

a(n) = Sum_{d|A052409(n)} binomial(A001055(n^(1/d)) + d - 1, d).

a(n) = a(A046523(n)). - Antti Karttunen, Nov 17 2019

Example

The a(36) = 12 multiset partitions:
  (2*2*3*3)    (6)*(2*3)  (6)*(6)  (36)
  (2*3)*(2*3)  (2*2*9)    (2*18)
               (2*3*6)    (3*12)
               (3*3*4)    (4*9)
                          (6*6)

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[With[{g=GCD@@FactorInteger[n][[All, 2]]}, Sum[Binomial[Length[facs[n^(1/d)]]+d-1, d], {d, Divisors[g]}]], {n, 100}]

Prog

(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
A320887(n) = if(1==n, n, my(r); sumdiv(A052409(n), d, binomial(A001055(sqrtnint(n, d)) + d - 1, d))); \\ Antti Karttunen, Nov 17 2019

Crossrefs

Cf. A001055, A001970, A046523, A050336, A052409, A279375, A294786, A295923, A320886, A320888, A320889.

Sequence in context: A324885 A046645 A284639 * A295923 A325806 A016470

Adjacent sequences: A320884 A320885 A320886 * A320888 A320889 A320890

Keyword

nonn

Author

Gus Wiseman, Oct 23 2018

Extensions

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

Status

approved