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

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

Formula

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

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 *)

Prog

(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));

Crossrefs

Cf. A181819, A258851.

Cf. also A351942.

Sequence in context: A113901 A062799 A063647 * A263653 A330328 A269427

Adjacent sequences: A353376 A353377 A353378 * A353380 A353381 A353382

Keyword

nonn

Author

Antti Karttunen, Apr 28 2022

Status

approved