A384423 The number of prime powers (not including 1) p^e that divide n such that e is unitarily coprime to the p-adic valuation of n.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 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, 4, 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, 4, 3, 2, 1, 3, 2, 2, 2

Offset: 1

Author

Amiram Eldar, May 28 2025

Keywords

• nonn
• easy

Comments

A number k is unitarily coprime to m if the largest divisor of k that is a unitary divisor of m is 1.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A047994, A077761, A085548, A321167.

Programs

Mathematica

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n];
ff[p_, e_] := uphi[e]; a[1] = 0; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

PARI

uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2]-1);}
a(n) = vecsum(apply(uphi, factor(n)[, 2]));

Formula

Additive with a(p^e) = uphi(e), where uphi is the unitary totient function (A047994).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B - C + D), where B is Mertens's constant (A077761), C = Sum_{p prime} 1/p^2 (A085548), and D = Sum_{p prime, e>=2} (1-1/p)*A047994(e)/p^e = 0.74335242036929441969... .