A384420 The number of exponentially squarefree prime powers (not including 1) that divide n.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Amiram Eldar, May 28 2025

Keywords

• nonn
• easy

Comments

The number of terms in A384419 that are larger than 1 and divide n.

Links

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

Crossrefs

Cf. A070321, A077761, A378085, A384419, A384421.

Programs

Mathematica

s[n_] := Module[{k = n}, While[! SquareFreeQ[k], k--]; k]; f[p_, e_] := s[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

s(n) = while(!issquarefree(n), n--); n;
a(n) = vecsum(apply(s, factor(n)[, 2]));

Formula

Additive with a(p^e) = A070321(e).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), and C = Sum_{p prime, e>=2} A378085(e-1)/p^e = 0.72770645470600638249... .