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

1,6

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).

Links

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

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)

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.

Similar sequences: A001222, A349258, A349281.

Cf. A083342, A179119.

Sequence in context: A174532 A089242 A349258 * A185894 A376305 A214180

Adjacent sequences: A349323 A349324 A349325 * A349327 A349328 A349329

Keyword

nonn,easy

Author

Amiram Eldar, Nov 15 2021

Status

approved