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%I #20 Jan 14 2021 17:06:39
%S 1,1,2,1,3,2,5,1,4,3,7,2,11,5,6,1,15,4,22,3,10,7,30,2,9,11,8,5,42,6,
%T 56,1,14,15,15,4,77,22,22,3,101,10,135,7,12,30,176,2,25,9,30,11,231,8,
%U 21,5,44,42,297,6,385,56,20,1,33,14,490,15,60,15,627,4
%N Number of twice-partitions whose domain is the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H Alois P. Heinz, <a href="/A299200/b299200.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(prime(n)) = A000041(n).
%e The a(15) = 6 twice-partitions: (3)(2), (3)(11), (21)(2), (21)(11), (111)(2), (111)(11).
%p with(numtheory): with(combinat):
%p a:= n-> mul(numbpart(pi(i[1]))^i[2], i=ifactors(n)[2]):
%p seq(a(n), n=1..82); # _Alois P. Heinz_, Jan 14 2021
%t Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsP[PrimePi[p]]^k],{n,100}]
%o (PARI) a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = numbpart(primepi(f[k, 1]));); factorback(f);} \\ _Michel Marcus_, Feb 26 2018
%Y Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299201, A299202, A299203.
%Y Cf. A000720, A003964.
%K nonn,look,mult
%O 1,3
%A _Gus Wiseman_, Feb 05 2018
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