A349258 a(n) is the number of prime powers (not including 1) that are infinitary divisors of n.

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

Offset: 1

Author

Amiram Eldar, Nov 12 2021

Keywords

• nonn
• easy

Comments

The total number of prime powers (not including 1) that divide n is A001222(n).

For each n, all the prime powers that are infinitary divisors of n are "Fermi-Dirac primes" (A050376).

Examples

			12 has 4 infinitary divisors, 1, 3, 4 and 12. Two of these divisors, 3 and 4 = 2^2 are prime powers. Therefore a(12) = 2.
		

Links

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

Crossrefs

Cf. A000120, A000961, A001222, A036537, A037445, A050376, A077609, A077761, A246655.

Programs

Mathematica

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

PARI

A349258(n) = if(1==n,0,vecsum(apply(x->(2^hammingweight(x))-1,factor(n)[,2]))); \\ Antti Karttunen, Nov 12 2021

Formula

Additive with a(p^e) = 2^A000120(e) - 1.

a(n) <= A001222(n), with equality if and only if n is in A036537.

a(n) <= A037445(n) - 1, with equality if and only if n is a prime power (including 1, A000961).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.28135949730844648114..., where f(x) = -(x+1) + (1-x) * Product_{k>=0} (1 + 2*x^(2^k)). - Amiram Eldar, Sep 29 2023

Extensions

Wrong comment removed by Amiram Eldar, Sep 22 2023