A383960 -id:A383960 - cp's OEIS frontend

A383959 The number of prime powers p^e having the property that e is a unitary 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

Amiram Eldar, May 16 2025

First differs from A238949 at n = 64.	
First differs from A383960 at n = 256.		
Also, the number of prime powers p^e having the property that e is a squarefree divisor of the p-adic valuation of n.

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

Cf. A001221, A034444, A077761, A085548, A238949, A278908, A361255, A383863, A383960.

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

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

Additive with a(p^e) = A034444(e) = 2^A001221(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)*A034444(e)/p^e = 0.92341081050532387352... .

cp's OEIS Frontend

A383959 The number of prime powers p^e having the property that e is a unitary divisor of the p-adic valuation of n.

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