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%I A361810 #9 Mar 27 2023 03:38:17
%S A361810 1,2,3,4,5,6,7,10,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,30,25,
%T A361810 26,30,28,29,30,31,34,33,34,35,36,37,38,39,50,41,42,43,44,45,46,47,48,
%U A361810 49,50,51,52,53,60,55,70,57,58,59,60,61,62,63,68,65,66,67,68
%N A361810 a(n) is the sum of divisors of n that are both infinitary and exponential.
%C A361810 The number of these divisors is A359411(n).
%C A361810 The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.
%H A361810 Amiram Eldar, <a href="/A361810/b361810.txt">Table of n, a(n) for n = 1..10000</a>
%F A361810 Multiplicative with a(p^e) = Sum_{d|e, bitor(d, e) == e} p^d.
%F A361810 a(n) >= n, with equality if and only if n is in A138302.
%F A361810 limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
%F A361810 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1 - 1/p)*(1 + Sum_{e>=1} Sum_{d|e, bitor(d, e) == e} p^(d-2*e))) = 0.51015879911178031024... .
%e A361810 a(8) = 10 since 8 has 2 divisors that are both infinitary and exponential, 2 and 8, and 2 + 8 = 10.
%t A361810 f[p_, e_] := DivisorSum[e, p^# &, BitOr[#, e] == e &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A361810 (PARI) s(p,e) = sumdiv(e, d, p^d*(bitor(d, e) == e));
%o A361810 a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); }
%Y A361810 Cf. A002110, A049417, A051377, A082020, A115964, A138302, A359411.
%K A361810 nonn,easy,mult
%O A361810 1,2
%A A361810 _Amiram Eldar_, Mar 25 2023

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