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A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

%I A366764 #7 Oct 21 2023 11:37:43
%S A366764 1,3,4,3,6,12,8,11,4,18,12,12,14,24,24,27,18,12,20,18,32,36,24,44,6,
%T A366764 42,31,24,30,72,32,59,48,54,48,12,38,60,56,66,42,96,44,36,24,72,48,
%U A366764 108,8,18,72,42,54,93,72,88,80,90,60,72,62,96,32,123,84,144,68
%N A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.
%C A366764 The sum of terms of A337050 that divide n.
%C A366764 The number of these divisors is A366763(n), and the largest of them is A366765(n).
%H A366764 Amiram Eldar, <a href="/A366764/b366764.txt">Table of n, a(n) for n = 1..10000</a>
%F A366764 Multiplicative with a(p) = p + 1, and a(p^e) = (p^(e+1) - 1)/(p - 1) - p^2 for e >= 2.
%F A366764 a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
%F A366764 a(n) >= A034448(n), with equality if and only if n is cubefree (A004709).
%F A366764 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(3*s-3)).
%F A366764 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3-p)) = 1.231291... (A065487).
%t A366764 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p^2; f[p_, 1] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366764 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, f[i, 1] + 1, (f[i, 1]^(f[i,2] + 1) - 1)/(f[i, 1] - 1) - f[i, 1]^2));}
%Y A366764 Cf. A000203, A004709, A005117, A034448, A065487, A337050, A366763, A366765.
%K A366764 nonn,easy,mult
%O A366764 1,2
%A A366764 _Amiram Eldar_, Oct 21 2023