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URL: https://oeis.org/A384422

⇱ A384422 - OEIS

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A384422
The number of prime powers (not including 1) p^e that divide n such that e is coprime to the p-adic valuation of n.
2
0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 4, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2
OFFSET
1,6
LINKS
FORMULA
Additive with a(p^e) = phi(e), where phi is the Euler totient function (A000010).
Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} f(1/p) = 0.24136815875213146317..., and f(x) = -x + (1-x)*x*Sum_{k>=1} mu(k)*x^(k-1)/(1-x^k)^2.
MATHEMATICA
f[p_, e_] := EulerPhi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(eulerphi, factor(n)[, 2]));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, May 28 2025
STATUS
approved