A362854 - OEIS

%I #7 May 05 2023 12:54:13

%S 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 26,30,28,29,30,31,34,33,34,35,36,37,38,39,50,41,42,43,44,45,46,47,54,

%U 49,50,51,52,53,60,55,70,57,58,59,60,61,62,63,70,65,66,67,68

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

%C The number of these divisors is A362852(n).

%C The indices of records of a(n)/n are the primorials (A002110) cubed, i.e., 1 and the terms of A115964.

%H Amiram Eldar, <a href="/A362854/b362854.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 a(n) >= n, with equality if and only if n is cubefree (A004709).

%F limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).

%F 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 a(8) = 10 since 8 has 2 divisors that are both bi-unitary and exponential, 2 and 8, and 2 + 8 = 10.

%t 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 (PARI) s(p, e) = sumdiv(e, d, p^d*(2*d != e));

%o a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2]));}

%Y Cf. A002110, A004709, A051377, A082020, A115964, A188999, A222266, A322791, A361810, A362852.

%K nonn,mult

%O 1,2

%A _Amiram Eldar_, May 05 2023