A357978 - OEIS

%I #13 Oct 04 2024 08:51:25

%S 1,2,2,4,3,4,3,8,4,6,5,8,7,6,6,16,11,8,13,12,6,10,19,16,9,14,8,12,29,

%T 12,37,32,10,22,9,16,47,26,14,24,61,12,79,20,12,38,103,32,9,18,22,28,

%U 131,16,15,24,26,58,163,24,199,74,12,64,21,20,251,44,38

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

%C In the definition, taking A000009(k) instead of prime(A000009(k)) gives A357982.

%e We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(1) * prime(1)^2 * prime(2) = 24.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];

%t Array[mtf[PartitionsQ],100]

%o (PARI) f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009

%o a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(f9(primepi(f[k,1])))); factorback(f); \\ _Michel Marcus_, Oct 25 2022

%Y The non-strict version is A357977.

%Y Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.

%Y A000040 lists the primes.

%Y A056239 adds up prime indices, row-sums of A112798.

%Y Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357979, A357983.

%K nonn,mult

%O 1,2

%A _Gus Wiseman_, Oct 24 2022