A368469 a(n) is the sum of exponentially odd divisors of the n-th exponentially odd number.

1, 3, 4, 6, 12, 8, 11, 18, 12, 14, 24, 24, 18, 20, 32, 36, 24, 44, 42, 31, 30, 72, 32, 43, 48, 54, 48, 38, 60, 56, 66, 42, 96, 44, 72, 48, 72, 54, 93, 72, 88, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 132, 90, 112

Offset

1,2

Links

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

Formula

a(n) = A033634(A268335(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^6/(1080*d^2)) * Product_{p prime} (1 - 1/p^2 - 2/p^4 + 2/p^5 + 1/p^6 - 1/p^7) = 0.98977643304712560492, and d = A065463 is the density of the exponentially odd numbers.

Mathematica

f[p_, e_] := 1 + (p^If[OddQ[e], e+2, e+1] - p)/(p^2 - 1); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]
(* or *)
f[p_, e_] := If[OddQ[e], 1 + (p^(e + 2) - p)/(p^2 - 1), 0]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 0 &]

Prog

(PARI) lista(kmax) = {my(f, s, p, e); for(k = 1, kmax, f = factor(k); s = prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(e%2, 1 + (p^(e+2) - p)/(p^2 - 1), 0)); if(s > 0, print1(s, ", "))); }

Crossrefs

Cf. A033634, A065463, A268335, A368468.

Sequence in context: A199970 A283738 A366535 * A374457 A366439 A062822

Adjacent sequences: A368466 A368467 A368468 * A368470 A368471 A368472

Keyword

nonn,easy

Author

Amiram Eldar, Dec 26 2023

Status

approved