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%I #8 Sep 02 2023 04:32:11
%S 1,1,1,3,1,1,1,3,4,1,1,3,1,1,1,11,1,4,1,3,1,1,1,3,6,1,4,3,1,1,1,11,1,
%T 1,1,12,1,1,1,3,1,1,1,3,4,1,1,11,8,6,1,3,1,4,1,3,1,1,1,3,1,1,4,43,1,1,
%U 1,3,1,1,1,12,1,1,6,3,1,1,1,11,31,1,1,3,1
%N The sum of exponentially odd divisors of the largest square dividing n.
%C The number of these divisors is A365333(n).
%H Amiram Eldar, <a href="/A365334/b365334.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A033634(A008833(n)).
%F a(n) = 1 if and only if n is squarefree (A005117).
%F Multiplicative with a(p^e) = 1 + (p^(e + 1 - (e mod 2)) - 1)/(p^2 - 1).
%F Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(2*s-1)).
%t f[p_, e_] := (p^(e + 1 - Mod[e, 2]) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2] + 1 - f[i,2]%2) - f[i,1])/(f[i,1]^2 - 1) + 1);}
%Y Cf. A008833, A005117, A033634, A365333.
%K nonn,easy,mult
%O 1,4
%A _Amiram Eldar_, Sep 01 2023
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