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A365347 The sum of divisors of the smallest number whose square is divisible by n.

%I A365347 #9 Sep 03 2023 00:03:16
%S A365347 1,3,4,3,6,12,8,7,4,18,12,12,14,24,24,7,18,12,20,18,32,36,24,28,6,42,
%T A365347 13,24,30,72,32,15,48,54,48,12,38,60,56,42,42,96,44,36,24,72,48,28,8,
%U A365347 18,72,42,54,39,72,56,80,90,60,72,62,96,32,15,84,144,68,54
%N A365347 The sum of divisors of the smallest number whose square is divisible by n.
%C A365347 The number of divisors of the smallest number whose square is divisible by n is A322483(n).
%C A365347 The sum of divisors of the smallest square divisible by n is A365346(n).
%H A365347 Amiram Eldar, <a href="/A365347/b365347.txt">Table of n, a(n) for n = 1..10000</a>
%F A365347 a(n) = A000203(A019554(n)).
%F A365347 Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).
%F A365347 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .
%t A365347 f[p_, e_] := (p^((e + Mod[e, 2])/2 + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365347 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^((f[i,2] + f[i,2]%2)/2 + 1) - 1)/(f[i,1] - 1));}
%o A365347 (PARI) a(n) = sigma(n/core(n, 1)[2]); \\ _Michel Marcus_, Sep 02 2023
%Y A365347 Cf. A000203, A019554, A322483, A365346.
%Y A365347 Cf. A002117, A065465.
%K A365347 nonn,easy,mult
%O A365347 1,2
%A A365347 _Amiram Eldar_, Sep 02 2023