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

1,6

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

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... .

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]

Prog

(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]));

Crossrefs

Cf. A047994, A077761, A085548, A321167.

Sequence in context: A008618 A339368 A225572 * A318471 A161234 A161058

Adjacent sequences: A384420 A384421 A384422 * A384424 A384425 A384426

Keyword

nonn,easy

Author

Amiram Eldar, May 28 2025

Status

approved