A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 3, 2, 1, 3, 1, 5, 2, 2, 2, 2, 1, 2, 2, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 4, 2, 4, 2, 2, 1, 3, 1, 2, 2, 5, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 3, 1, 4, 3, 2, 1, 3, 2, 2, 2, 4, 1, 3, 2, 2, 2, 2, 2, 6, 1, 2, 2, 2, 1, 3, 1, 4, 3

Offset: 1

Author

Amiram Eldar, Nov 15 2021

Keywords

• nonn
• easy

Comments

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

The least number k such that a(k) = m is A122756(m).

Examples

			12 has 4 bi-unitary 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. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.

Similar sequences: A001222, A349258, A349281.

Cf. A083342, A179119.

Programs

Mathematica

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

PARI

a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ Amiram Eldar, Sep 29 2023

Formula

Additive with a(p^e) = e if e is odd, and e-1 if e is even.

a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).

a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).

a(n) = A001222(n) - A162641(n). - Amiram Eldar, May 18 2023

From Amiram Eldar, Sep 29 2023: (Start)

a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)