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%I #15 Nov 17 2019 15:59:24
%S 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,
%T 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,
%U 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
%N Number of multiset partitions of factorizations of n into factors > 1 such that all the parts have the same product.
%H Antti Karttunen, <a href="/A320887/b320887.txt">Table of n, a(n) for n = 1..10080</a>
%H Antti Karttunen, <a href="/A320887/a320887.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F a(n) = Sum_{d|A052409(n)} binomial(A001055(n^(1/d)) + d - 1, d).
%F a(n) = a(A046523(n)). - _Antti Karttunen_, Nov 17 2019
%e The a(36) = 12 multiset partitions:
%e (2*2*3*3) (6)*(2*3) (6)*(6) (36)
%e (2*3)*(2*3) (2*2*9) (2*18)
%e (2*3*6) (3*12)
%e (3*3*4) (4*9)
%e (6*6)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[With[{g=GCD@@FactorInteger[n][[All,2]]},Sum[Binomial[Length[facs[n^(1/d)]]+d-1,d],{d,Divisors[g]}]],{n,100}]
%o (PARI)
%o 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));
%o A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
%o 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
%Y Cf. A001055, A001970, A046523, A050336, A052409, A279375, A294786, A295923, A320886, A320888, A320889.
%K nonn
%O 1,4
%A _Gus Wiseman_, Oct 23 2018
%E Data section extended up to term a(105) by _Antti Karttunen_, Nov 17 2019
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