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A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).

%I A366539 #11 Apr 26 2025 21:35:22
%S A366539 1,3,4,5,6,12,8,10,18,12,20,14,24,24,17,18,30,20,30,32,36,24,26,42,40,
%T A366539 30,72,32,48,54,48,50,38,60,56,42,96,44,60,60,72,48,68,50,78,72,70,54,
%U A366539 72,80,90,60,120,62,96,80,84,144,68,90,96,144,72,74,114,104
%N A366539 The sum of unitary divisors of the exponentially 2^n-numbers (A138302).
%C A366539 Also the sum of infinitary divisors of the terms of A138302, since A138302 is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.
%H A366539 Amiram Eldar, <a href="/A366539/b366539.txt">Table of n, a(n) for n = 1..10000</a>
%F A366539 a(n) = A034448(A138302(n)).
%F A366539 a(n) = A049417(A138302(n)).
%F A366539 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} f(1/p) = 1.58107339851877782285..., d = A271727 is the asymptotic density of A138302, and f(x) = 1 + x^2 + 2 * Sum_{k>=2} (x^(2^k)-x^(2^k+1)).
%F A366539 The asymptotic mean of the unitary abundancy index of A138302: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A138302(k) = c * d = 1.37948208055913856387... .
%t A366539 s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, # == 2^IntegerExponent[#, 2] &], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]
%o A366539 (PARI) lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], is = 1); for(i = 1, #e, if(e[i] >> valuation(e[i], 2) > 1, is = 0; break)); if(is, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));
%Y A366539 Cf. A034448, A049417, A077609, A077610, A138302, A271727, A366538.
%Y A366539 Similar sequences: A034676, A366535, A366537.
%K A366539 nonn,easy
%O A366539 1,2
%A A366539 _Amiram Eldar_, Oct 12 2023