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A368469
a(n) is the sum of exponentially odd divisors of the n-th exponentially odd number.
2
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
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
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Dec 26 2023
STATUS
approved