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A320887 Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 8 09:16 EDT 2020. Contains 336293 sequences. (Running on oeis4.)