A353379 Primepi-based variant of the arithmetic derivative applied to the prime shadow of n.

0, 1, 1, 2, 1, 4, 1, 3, 2, 4, 1, 7, 1, 4, 4, 4, 1, 7, 1, 7, 4, 4, 1, 11, 2, 4, 3, 7, 1, 12, 1, 5, 4, 4, 4, 12, 1, 4, 4, 11, 1, 12, 1, 7, 7, 4, 1, 15, 2, 7, 4, 7, 1, 11, 4, 11, 4, 4, 1, 20, 1, 4, 7, 6, 4, 12, 1, 7, 4, 12, 1, 19, 1, 4, 7, 7, 4, 12, 1, 15, 4, 4, 1, 20, 4, 4, 4, 11, 1, 20, 4, 7, 4, 4, 4, 21, 1, 7, 7

Offset: 1

Author

Antti Karttunen, Apr 28 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A181819, A258851.

Cf. also A351942.

Programs

Maple

a:= n-> (m-> m*add(i[2]*numtheory[pi](i[1])/i[1], i=ifactors(m)[2]))
        (mul(ithprime(i[2]), i=ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Apr 28 2022

Mathematica

a[n_] := If[n == 1, 0, #*Sum[i[[2]]*PrimePi[i[[1]]]/i[[1]], {i, FactorInteger[#]}]]&[Product[Prime[i[[2]]], {i, FactorInteger[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 29 2025, after Alois P. Heinz *)

PARI

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A258851(n) = (n*sum(i=1, #n=factor(n)~, n[2, i]*primepi(n[1, i])/n[1, i])); \\ From A258851
A353379(n) = A258851(A181819(n));

Formula

a(n) = A258851(A181819(n)).