cp's OEIS Frontend

A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.

%I A357980 #13 Oct 04 2024 08:51:37
%S A357980 1,1,2,1,3,2,3,1,4,3,5,2,5,3,6,1,7,4,7,3,6,5,7,2,9,5,8,3,7,6,11,1,10,
%T A357980 7,9,4,11,7,10,3,13,6,13,5,12,7,13,2,9,9,14,5,13,8,15,3,14,7,17,6,17,
%U A357980 11,12,1,15,10,19,7,14,9,19,4,19,11,18,7,15,10
%N A357980 Replace prime(k) with prime(A000720(k)) in the prime factorization of n, assuming prime(0) = 1.
%C A357980 In the definition, taking A000720(k) in place of prime(A000720(k)) gives A357984.
%e A357980 We have 90 = prime(1) * prime(2)^2 * prime(3), so a(90) = prime(0) * prime(1)^2 * prime(2) = 12.
%t A357980 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357980 mtf[f_][n_]:=Product[If[f[i]==0,1,Prime[f[i]]],{i,primeMS[n]}];
%t A357980 Array[mtf[PrimePi],100]
%o A357980 (PARI) myprime(n) = if (n==0, 1, prime(n));
%o A357980 a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = myprime(primepi(primepi(f[k,1])))); factorback(f); \\ _Michel Marcus_, Oct 25 2022
%Y A357980 Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
%Y A357980 The version for p instead of pi is A357977, strict A357978.
%Y A357980 The triangular version is A357984.
%Y A357980 A000040 lists the prime numbers.
%Y A357980 A000720 = PrimePi.
%Y A357980 A056239 adds up prime indices, row-sums of A112798.
%Y A357980 Cf. A063834, A066207, A076610, A215366, A296150, A357975, A357983.
%K A357980 nonn,mult
%O A357980 1,3
%A A357980 _Gus Wiseman_, Oct 24 2022