A349281 a(n) is the number of prime powers (not including 1) that are (1+e)-divisors of n.

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

Offset: 1

Author

Amiram Eldar, Nov 13 2021

Keywords

• nonn
• easy

Comments

(1+e)-divisors are defined in A049599.

First differs from A106490 at n = 64.

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

If p|n and p^e is the highest power of p that divides n, then the powers of p that are (1+e)-divisors of n are of the form p^d where d|e.

Examples

			8 has 3 (1+e)-divisors, 1, 2 and 8. Two of these divisors, 2 and 8 = 2^3 are prime powers. Therefore, a(8) = 2.
		

Links

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

Crossrefs

Cf. A000005, A000961, A001222, A046099, A049599, A077761, A106490, A246655, A349258.

Programs

Mathematica

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

PARI

A349281(n) = vecsum(apply(e->numdiv(e),factor(n)[,2])); \\ Antti Karttunen, Nov 13 2021

Formula

Additive with a(p^e) = A000005(e).

a(n) <= A001222(n), with equality if and only if n is cubefree (A046099).

a(n) <= A049599(n)-1, with equality if and only if n is a prime power (including 1, A000961).

Sum_{k=1..n} a(n) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{p prime} f(1/p) = 0.51780076119050171903..., where f(x) = -x + (1-x) * Sum_{k>=1} x^k/(1-x^k). - Amiram Eldar, Sep 29 2023