A384420 -id:A384420 - cp's OEIS frontend

A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

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

Offset: 1

Amiram Eldar, May 28 2025

First differs from A125029 at n = 64.
A number k unitarily divides n if k|n and gcd(k, n/k) = 1.
The number of unitary divisors of n that are larger than 1 and are terms in A384419.

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

Cf. A008966, A125029, A383959, A384419, A384420.

f[p_, e_] := If[SquareFreeQ[e], 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(issquarefree, factor(n)[, 2]));

Additive with a(p^e) = A008966(e).

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)*A008966(e)/p^e = 0.40780808646244052181... .

cp's OEIS Frontend

A384421 The number of exponentially squarefree prime powers (not including 1) that unitarily divide n.

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