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

Author

Amiram Eldar, Dec 26 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A033634, A065463, A268335, A368468.

Programs

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 &]

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, ", ")));}

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.