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a(n) is the sum of exponentially odd divisors of the n-th exponentially odd number.
2

%I #7 Dec 27 2023 01:21:49

%S 1,3,4,6,12,8,11,18,12,14,24,24,18,20,32,36,24,44,42,31,30,72,32,43,

%T 48,54,48,38,60,56,66,42,96,44,72,48,72,54,93,72,88,80,90,60,62,96,84,

%U 144,68,96,144,72,74,114,96,168,80,126,84,108,132,120,132,90,112

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

%H Amiram Eldar, <a href="/A368469/b368469.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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.

%t 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] &]

%t (* or *)

%t 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 &]

%o (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, ", ")));}

%Y Cf. A033634, A065463, A268335, A368468.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Dec 26 2023