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

1,4

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.

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A076755 A317751 A106490 * A372502 A345222 A327399

Adjacent sequences: A349278 A349279 A349280 * A349282 A349283 A349284

Keyword

nonn,easy

Author

Amiram Eldar, Nov 13 2021

Status

approved