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%I #8 May 28 2023 20:50:24
%S 1,2,3,4,5,6,7,14,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,42,25,
%T 26,39,28,29,30,31,50,33,34,35,36,37,38,39,70,41,42,43,44,45,46,47,48,
%U 49,50,51,52,53,78,55,98,57,58,59,60,61,62,63,84,65,66,67,68
%N a(n) is the sum of divisors of n that are both coreful and infinitary.
%C First differs from A363334 at n = 16.
%C The number of these divisors is A363329(n).
%H Amiram Eldar, <a href="/A363331/b363331.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = (Product_{k>=0} ((p^(2^k*(b(k)+1)) - 1)/(p^(2^k) - 1)) - 1, where e = Sum_{k >= 0} b(k) * 2^k is the binary representation of e.
%F a(n) >= n, with equality if and only if n is in A138302.
%F a(n) >= A361810(n), with equality if and only if n is in A138302.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p) * Sum_{k>=1} a(p^k)/p^(2*k)) = 0.53906337497505398777... .
%e a(8) = 14 since 8 has 3 divisors that are both infinitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.
%t f[p_, e_] := Times @@ (1 + Flatten[p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], _?(# == 1 &)]))]) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(f[i, 2]); prod(k = 1, #b, if(b[k], f[i, 1]^(2^(#b - k)) + 1, 1)) - 1);}
%Y Cf. A049417, A057723, A077609, A138302, A361810, A363329, A363334.
%K nonn,mult
%O 1,2
%A _Amiram Eldar_, May 28 2023
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