cp's OEIS Frontend

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

%I A357982 #19 Oct 04 2024 08:51:16
%S A357982 1,1,1,1,2,1,2,1,1,2,3,1,4,2,2,1,5,1,6,2,2,3,8,1,4,4,1,2,10,2,12,1,3,
%T A357982 5,4,1,15,6,4,2,18,2,22,3,2,8,27,1,4,4,5,4,32,1,6,2,6,10,38,2,46,12,2,
%U A357982 1,8,3,54,5,8,4,64,1,76,15,4,6,6,4,89,2,1
%N A357982 Replace prime(k) with A000009(k) in the prime factorization of n.
%C A357982 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. This sequence gives the number of ways to choose a strict partition of each prime index of n.
%C A357982 The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - _Ivan N. Ianakiev_, Oct 27 2022
%e A357982 The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).
%t A357982 Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]
%o A357982 (PARI) f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
%o A357982 a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ _Michel Marcus_, Oct 26 2022
%Y A357982 Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
%Y A357982 The non-strict version is A299200.
%Y A357982 A horizontal version is A357978, non-strict A357977.
%Y A357982 A000040 lists the primes.
%Y A357982 A056239 adds up prime indices, row-sums of A112798.
%Y A357982 Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
%Y A357982 Cf. A003586, A059485.
%K A357982 nonn,mult
%O A357982 1,5
%A A357982 _Gus Wiseman_, Oct 25 2022