A076598 Sum of squares of divisors d of n such that d or n/d is odd.
1, 5, 10, 17, 26, 50, 50, 65, 91, 130, 122, 170, 170, 250, 260, 257, 290, 455, 362, 442, 500, 610, 530, 650, 651, 850, 820, 850, 842, 1300, 962, 1025, 1220, 1450, 1300, 1547, 1370, 1810, 1700, 1690, 1682, 2500, 1850, 2074, 2366, 2650, 2210, 2570, 2451, 3255
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
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 *)
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PARI
a(n) = sumdiv(n, d, if ((d % 2) || (n/d % 2), d^2)); \\ Michel Marcus, Oct 30 2022
Formula
Multiplicative with a(2^e) = 4^e+1, a(p^e) = (p^(2*e+2)-1)/(p^2-1) for an odd prime p.
G.f.: Sum_{m>0} m^2*x^m*(1+2*x^m+3*x^(2*m))/(1+x^(2*m))/(1+x^m).
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).
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.
a(n) = sigma_2(n)-4*sigma_2(n/4) if n mod 4=0, otherwise a(n) = sigma_2(n).
Sum_{k=1..n} a(k) ~ c * n^3, where c = 5*zeta(3)/16 = 0.375642... . - Amiram Eldar, Oct 30 2022