A392016 The sum of the exponential unitary divisors of the exponentially-2^n numbers.

1, 2, 3, 6, 5, 6, 7, 12, 10, 11, 18, 13, 14, 15, 18, 17, 24, 19, 30, 21, 22, 23, 30, 26, 42, 29, 30, 31, 33, 34, 35, 72, 37, 38, 39, 41, 42, 43, 66, 60, 46, 47, 54, 56, 60, 51, 78, 53, 55, 57, 58, 59, 90, 61, 62, 84, 65, 66, 67, 102, 69, 70, 71, 73, 74, 90, 114

Offset

1,2

Comments

The number of these divisors is A392014(n).

Links

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

Formula

a(n) = A322857(A138302(n)).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (1/d^2) * Product_{p prime} f(1/p) = 1.27507469612272720041..., where d = A271727, and f(x) = (1-x) * (1 - x^5 + Sum_{k>=3} (x^(2^k-1) - x^(2^k+1))).

Mathematica

f[p_, e_] := Module[{v = IntegerExponent[e, 2]}, If[e == 2^v, If[v == 0, p, p + p^(2^v)], 0]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100] , # > 0 &]

Prog

(PARI) s(n) = {my(f = factor(n)); prod(i = 1, #f~, my(v = valuation(f[i, 2], 2)); if(f[i, 2] == 1 << v, f[i, 1] + if(v > 0, f[i, 1]^(1 << v), 0))); }
list(lim) = select(x -> x > 0, vector(lim, i, s(i)));

Crossrefs

Cf. A138302, A271727, A322857, A361255, A392014, A392015.

Sequence in context: A361175 A051377 A392012 * A369319 A336465 A340774

Adjacent sequences: A392013 A392014 A392015 * A392017 A392018 A392019

Keyword

nonn,easy

Author

Amiram Eldar, Dec 27 2025

Status

approved