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

%I A349326 #24 Aug 17 2025 02:10:32
%S A349326 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,
%T A349326 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,
%U A349326 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
%N A349326 a(n) is the number of prime powers (not including 1) that are bi-unitary divisors of n.
%C A349326 The total number of prime powers (not including 1) that divide n is A001222(n).
%C A349326 The least number k such that a(k) = m is A122756(m).
%H A349326 Amiram Eldar, <a href="/A349326/b349326.txt">Table of n, a(n) for n = 1..10000</a>
%F A349326 Additive with a(p^e) = e if e is odd, and e-1 if e is even.
%F A349326 a(n) <= A001222(n), with equality if and only if n is an exponentially odd number (A268335).
%F A349326 a(n) <= A286324(n) - 1, with equality if and only if n is a prime power (including 1, A000961).
%F A349326 a(n) = A001222(n) - A162641(n). - _Amiram Eldar_, May 18 2023
%F A349326 From _Amiram Eldar_, Sep 29 2023: (Start)
%F A349326 a(n) = A001222(A350390(n)) (the number of prime factors of the largest exponentially odd number dividing n, counted with multiplicity).
%F A349326 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B_2 - C), where B_2 = A083342 and C = A179119. (End)
%e A349326 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.
%t A349326 f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A349326 (PARI) a(n) = vecsum(apply(x -> if(x%2, x, x-1), factor(n)[, 2])); \\ _Amiram Eldar_, Sep 29 2023
%Y A349326 Cf. A000961, A001222, A122756, A162641, A222266, A246655, A286324, A268335, A350390.
%Y A349326 Similar sequences: A001222, A349258, A349281.
%Y A349326 Cf. A083342, A179119.
%K A349326 nonn,easy
%O A349326 1,6
%A A349326 _Amiram Eldar_, Nov 15 2021