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A368172 The number of divisors of the largest cubefull exponentially odd divisor of n (A368170).

%I A368172 #9 Dec 16 2023 09:01:55
%S A368172 1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,4,1,1,1,1,6,1,1,
%T A368172 1,1,1,1,1,4,1,1,1,1,1,1,1,4,1,1,1,1,1,4,1,4,1,1,1,1,1,1,1,6,1,1,1,1,
%U A368172 1,1,1,4,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1
%N A368172 The number of divisors of the largest cubefull exponentially odd divisor of n (A368170).
%C A368172 First differs from A365487 at n = 32.
%H A368172 Amiram Eldar, <a href="/A368172/b368172.txt">Table of n, a(n) for n = 1..10000</a>
%F A368172 a(n) = A000005(A368170(n)).
%F A368172 Multiplicative with a(p^e) = 1 if e <= 2, a(p^e) = e+1 if e is odd and e > 1, and a(p^e) = e otherwise.
%F A368172 a(n) >= 1, with equality if and only if n is cubefree (A004709).
%F A368172 a(n) <= A000005(n), with equality if and only if n is in A335988.
%F A368172 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 3/p^3 - 1/p^5) = 1.69824776889117043774... .
%t A368172 f[p_, e_] := If[e <= 2, 1, If[EvenQ[e], e, e + 1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368172 (PARI) a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2] <= 2, 1, if(!(f[i,2]%2), f[i, 2], f[i, 2]+1)))};
%Y A368172 Cf. A000005, A013661, A004709, A286324, A365487, A368170.
%K A368172 nonn,easy,mult
%O A368172 1,8
%A A368172 _Amiram Eldar_, Dec 14 2023