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A362854 The sum of the divisors of n that are both bi-unitary and exponential.

%I A362854 #7 May 05 2023 12:54:13
%S A362854 1,2,3,4,5,6,7,10,9,10,11,12,13,14,15,18,17,18,19,20,21,22,23,30,25,
%T A362854 26,30,28,29,30,31,34,33,34,35,36,37,38,39,50,41,42,43,44,45,46,47,54,
%U A362854 49,50,51,52,53,60,55,70,57,58,59,60,61,62,63,70,65,66,67,68
%N A362854 The sum of the divisors of n that are both bi-unitary and exponential.
%C A362854 The number of these divisors is A362852(n).
%C A362854 The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.
%H A362854 Amiram Eldar, <a href="/A362854/b362854.txt">Table of n, a(n) for n = 1..10000</a>
%F A362854 Multiplicative with a(p^e) = Sum_{d|e} p^d if e is odd, and (Sum_{d|e} p^d) - p^(e/2) if e is even.
%F A362854 a(n) >= n, with equality if and only if n is cubefree (A004709).
%F A362854 limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
%F A362854 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, d != e/2}, p^(d-2*e))) = 0.5124353304539905... .
%e A362854 a(8) = 10 since 8 has 2 divisors that are both bi-unitary and exponential, 2 and 8, and 2 + 8 = 10.
%t A362854 f[p_, e_] := DivisorSum[e, p^# &] - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A362854 (PARI) s(p, e) = sumdiv(e, d, p^d*(2*d != e));
%o A362854 a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2]));}
%Y A362854 Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.
%K A362854 nonn,mult
%O A362854 1,2
%A A362854 _Amiram Eldar_, May 05 2023