A383960 The number of prime powers p^e having the property that e is an infinitary divisor of the p-adic valuation of n.

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

Offset: 1

Author

Amiram Eldar, May 16 2025

Keywords

• nonn
• easy

Comments

First differs from A238949 at n = 64.

First differs from A383959 at n = 256.

Links

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

Crossrefs

Cf. A037445, A085548, A238949, A383760, A383865, A383959.

Programs

Mathematica

f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 0; ff[p_, e_] := d[e]; a[n_] := Plus @@ ff @@@ FactorInteger[n]; Array[a, 100]

PARI

d(n) = vecprod(apply(x -> 1 << hammingweight(x), factor(n)[, 2]));
a(n) = vecsum(apply(x -> d(x), factor(n)[, 2]));

Formula

Additive with a(p^e) = A037445(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)*A037445(e)/p^e = 0.92752481299257205938... .