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

1,2

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

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... .

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]

Prog

(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]), ", ")));

Crossrefs

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

Similar sequences: A034676, A366535, A366537.

Sequence in context: A331107 A069184 A181549 * A366537 A241405 A322485

Adjacent sequences: A366536 A366537 A366538 * A366540 A366541 A366542

Keyword

nonn,easy

Author

Amiram Eldar, Oct 12 2023

Status

approved