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A366763 The number of divisors of n that have no exponent 2 in their prime factorization.

%I A366763 #12 Apr 20 2025 08:26:10
%S A366763 1,2,2,2,2,4,2,3,2,4,2,4,2,4,4,4,2,4,2,4,4,4,2,6,2,4,3,4,2,8,2,5,4,4,
%T A366763 4,4,2,4,4,6,2,8,2,4,4,4,2,8,2,4,4,4,2,6,4,6,4,4,2,8,2,4,4,6,4,8,2,4,
%U A366763 4,8,2,6,2,4,4,4,4,8,2,8,4,4,2,8,4,4,4
%N A366763 The number of divisors of n that have no exponent 2 in their prime factorization.
%C A366763 The number of terms of A337050 that divide n.
%C A366763 The sum of these divisors is A366764(n), and the largest of them is A366765(n).
%H A366763 Amiram Eldar, <a href="/A366763/b366763.txt">Table of n, a(n) for n = 1..10000</a>
%H A366763 Vaclav Kotesovec, <a href="/A366763/a366763.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A366763 Multiplicative with a(p^e) = max(e, 2);
%F A366763 a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
%F A366763 a(n) >= A034444(n), with equality if and only if n is cubefree (A004709).
%F A366763 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
%F A366763 From _Vaclav Kotesovec_, Apr 20 2025: (Start)
%F A366763 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)).
%F A366763 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
%F A366763 f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3) = A330596 = 0.74853525968236356464421504863791060164164034300532440451585279392592558689...,
%F A366763 f'(1) = f(1) * Sum_{p prime} (2*p-3)*log(p)/(p^3-p+1) = f(1) * 0.560697508735949606541137451100554649565120075155278833396722097786365686597...
%F A366763 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A366763 f[p_, e_] := Max[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366763 (PARI) a(n) = vecprod(apply(x -> max(x, 2), factor(n)[, 2]));
%Y A366763 Cf. A000005, A004709, A005117, A034444, A337050, A366764, A366765.
%Y A366763 Cf. A330596.
%K A366763 nonn,easy,mult
%O A366763 1,2
%A A366763 _Amiram Eldar_, Oct 21 2023