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

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, 30, 72, 32, 48, 54, 48, 50, 38, 60, 56, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 72, 80, 90, 60, 120, 62, 96, 80, 84, 144, 68, 90, 96, 144, 72, 74, 114, 104

Offset: 1

Author

Amiram Eldar, Oct 12 2023

Keywords

• nonn
• easy

Comments

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A034448, A049417, A077609, A077610, A138302, A271727, A366538.

Similar sequences: A034676, A366535, A366537.

Programs

Mathematica

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]

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]), ", ")));

Formula

a(n) = A034448(A138302(n)).

a(n) = A049417(A138302(n)).

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)).

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... .