cp's OEIS Frontend

A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.

%I A357977 #19 Oct 04 2024 08:51:33
%S A357977 1,2,3,4,5,6,11,8,9,10,17,12,31,22,15,16,47,18,79,20,33,34,113,24,25,
%T A357977 62,27,44,181,30,263,32,51,94,55,36,389,158,93,40,547,66,761,68,45,
%U A357977 226,1049,48,121,50,141,124,1453,54,85,88,237,362,1951,60,2659,526
%N A357977 Replace prime(k) with prime(A000041(k)) in the prime factorization of n.
%C A357977 In the definition, taking A000041(k) instead of prime(A000041(k)) gives A299200.
%e A357977 We have 35 = prime(3) * prime(4), so a(35) = prime(A000041(3)) * prime(A000041(4)) = prime(3) * prime(5) = 55.
%t A357977 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357977 mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
%t A357977 Array[mtf[PartitionsP],100]
%o A357977 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = prime(numbpart(primepi(f[k,1])))); factorback(f); \\ _Michel Marcus_, Oct 25 2022
%Y A357977 Applying the same transformation again gives A357979.
%Y A357977 The strict version is A357978.
%Y A357977 Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
%Y A357977 A000040 lists the primes.
%Y A357977 A056239 adds up prime indices, row-sums of A112798.
%Y A357977 Cf. A000041, A000720, A003964, A063834, A076610, A215366, A296150, A299201, A299202, A357975, A357983.
%K A357977 nonn,mult
%O A357977 1,2
%A A357977 _Gus Wiseman_, Oct 23 2022