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A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.

%I A363332 #10 Sep 24 2023 04:10:43
%S A363332 1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,5,1,1,
%T A363332 1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,5,1,1,1,1,
%U A363332 1,1,1,3,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1
%N A363332 a(n) is the number of divisors of n that are both coreful and bi-unitary.
%C A363332 For the definition of a coreful divisor see A307958, and for the definition of a bi-unitary divisor see A222266.
%C A363332 If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a coreful and an bi-unitary divisor, the exponent of the highest power of p dividing d is a number k >= 1 that is not equal to e/2.
%C A363332 All the terms are odd.
%H A363332 Amiram Eldar, <a href="/A363332/b363332.txt">Table of n, a(n) for n = 1..10000</a>
%F A363332 Multiplicative with a(p^e) = e - 1 + (e mod 2).
%F A363332 a(n) = 1 if and only if n is cubefree (A004709).
%F A363332 a(n) >= A362852(n), with equality if and only if n is cubefree.
%F A363332 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 2/(p^3-p)) = 1.48264570900305853294... .
%F A363332 Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 2/p^(3*s)). - _Amiram Eldar_, Sep 24 2023
%e A363332 a(8) = 3 since 8 has 4 divisors, 1, 2, 4 and 8, all are bi-unitary and 3 of them (2, 4 and 8) are also coreful.
%t A363332 f[p_, e_] := If[OddQ[e], e, e - 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 120]
%o A363332 (PARI) a(n)={my(e = factor(n)[,2]); prod(i=1, #e, e[i] - 1 + e[i] % 2);}
%Y A363332 Cf. A004709, A005361 (number of coreful divisors), A222266, A286324, A362852.
%K A363332 nonn,easy,mult
%O A363332 1,8
%A A363332 _Amiram Eldar_, May 28 2023