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A076598 Sum of squares of divisors d of n such that d or n/d is odd.

%I A076598 #17 Oct 30 2022 05:27:40
%S A076598 1,5,10,17,26,50,50,65,91,130,122,170,170,250,260,257,290,455,362,442,
%T A076598 500,610,530,650,651,850,820,850,842,1300,962,1025,1220,1450,1300,
%U A076598 1547,1370,1810,1700,1690,1682,2500,1850,2074,2366,2650,2210,2570,2451,3255
%N A076598 Sum of squares of divisors d of n such that d or n/d is odd.
%H A076598 Amiram Eldar, <a href="/A076598/b076598.txt">Table of n, a(n) for n = 1..10000</a>
%F A076598 Multiplicative with a(2^e) = 4^e+1, a(p^e) = (p^(2*e+2)-1)/(p^2-1) for an odd prime p.
%F A076598 G.f.: Sum_{m>0} m^2*x^m*(1+2*x^m+3*x^(2*m))/(1+x^(2*m))/(1+x^m).
%F A076598 More generally, if b(n, k) is sum of k-th powers of divisors d of n such that d or n/d is odd then b(n, k) = sigma_k(n)-2^k*sigma_k(n/4) if n mod 4=0, otherwise b(n, k) = sigma_k(n).
%F A076598 G.f. for b(n, k): Sum_{m>0} m^k*x^m*(1+x^m+x^(2*m)-(2^k-1)*x^(3*m))/(1-x^(4*m)). b(n, k) is multiplicative and b(2^e, k) = 2^(k*e)+1, b(p^e, k) = (p^(k*e+k)-1)/(p^k-1) for an odd prime p.
%F A076598 a(n) = sigma_2(n)-4*sigma_2(n/4) if n mod 4=0, otherwise a(n) = sigma_2(n).
%F A076598 Sum_{k=1..n} a(k) ~ c * n^3, where c = 5*zeta(3)/16 = 0.375642... . - _Amiram Eldar_, Oct 30 2022
%t A076598 f[2, e_] := 4^e+1 ; f[p_, e_] := (p^(2*e+2)-1)/(p^2-1) ; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 50] (* _Amiram Eldar_, Aug 01 2019 *)
%o A076598 (PARI) a(n) = sumdiv(n, d, if ((d % 2) || (n/d % 2), d^2)); \\ _Michel Marcus_, Oct 30 2022
%Y A076598 Cf. A002117, A069733, A069184, A001157, A050999, A076577.
%K A076598 mult,nonn
%O A076598 1,2
%A A076598 _Vladeta Jovovic_, Oct 20 2002