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A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

%I A368471 #7 Dec 27 2023 01:21:13
%S A368471 1,3,4,1,6,12,8,11,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,44,1,42,31,
%T A368471 8,30,72,32,43,48,54,48,1,38,60,56,66,42,96,44,12,6,72,48,4,1,3,72,14,
%U A368471 54,93,72,88,80,90,60,24,62,96,8,1,84,144,68,18,96,144
%N A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).
%H A368471 Amiram Eldar, <a href="/A368471/b368471.txt">Table of n, a(n) for n = 1..10000</a>
%F A368471 a(n) = A033634(A350389(n)).
%F A368471 Multiplicative with a(p^e) = (p^(e+2) - p)/(p^2 - 1) + 1 if e is odd and 1 otherwise.
%F A368471 a(n) >= 1, with equality if and only if n is a square (A000290).
%F A368471 a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
%F A368471 Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^6/1080) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.51287686448947428073... .
%t A368471 f[p_, e_] := If[OddQ[e], 1 + (p^(e + 2) - p)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368471 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, (f[i,1]^(f[i,2]+2) - f[i,1])/(f[i,1]^2 - 1) + 1, 1));}
%Y A368471 Cf. A000203, A000290, A005117, A033634, A268335, A350389, A368470.
%K A368471 nonn,easy,mult
%O A368471 1,2
%A A368471 _Amiram Eldar_, Dec 26 2023