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A366744 The sum of divisors of the least coreful infinitary divisor of n.

%I A366744 #9 Oct 21 2023 01:50:25
%S A366744 1,3,4,7,6,12,8,3,13,18,12,28,14,24,24,31,18,39,20,42,32,36,24,12,31,
%T A366744 42,4,56,30,72,32,3,48,54,48,91,38,60,56,18,42,96,44,84,78,72,48,124,
%U A366744 57,93,72,98,54,12,72,24,80,90,60,168,62,96,104,7,84,144,68
%N A366744 The sum of divisors of the least coreful infinitary divisor of n.
%C A366744 The number of divisors of the least coreful infinitary divisor of n is A366742(n).
%H A366744 Amiram Eldar, <a href="/A366744/b366744.txt">Table of n, a(n) for n = 1..10000</a>
%F A366744 a(n) = A000203(A365296(n)).
%F A366744 a(n) = A000203(n) if and only if n is in A138302.
%F A366744 Multiplicative with a(p^e) = (p^(A006519(e)+1) - 1)/(p - 1).
%F A366744 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/(p^2-1) + Sum_{e>=1} 1/p^f(e)) = 0.696427154..., where f(k) = 2*k - A006519(k) = A339597(k-1).
%t A366744 f[p_, e_] := (p^(2^IntegerExponent[e, 2]+1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366744 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(2^valuation(f[i, 2], 2)+1) - 1)/(f[i, 1] -1 ));}
%Y A366744 Cf. A000203, A006519, A138302, A339597, A365296, A366742, A366743.
%K A366744 nonn,easy,mult
%O A366744 1,2
%A A366744 _Amiram Eldar_, Oct 19 2023