A076577 -id:A076577 - cp's OEIS frontend

A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

1, -4, 0, 8, 16, -8, -48, -56, 0, 116, 256, 264, -32, -648, -1296, -1392, -352, 2040, 5200, 7368, 6112, -784, -13744, -29304, -39648, -33804, -1376, 60368, 139552, 205304, 210208, 103432, -146528, -521744, -928480, -1190000, -1069904, -339720, 1110864, 3146640, 5278624

Offset: 0

Paul D. Hanna, Sep 06 2012

sign

The number of contiguous signs seems to increase in proportion to the square-root of the number of terms.
Compare the g.f. to the Jacobi theta_4 series identity:
exp( Sum_{n>=1} -(sigma(2*n) - sigma(n))*x^n/n ) = 1 + 2*Sum_{n>=1} (-x)^(n^2).

			G.f.: A(x) = 1 - 4*x + 8*x^3 + 16*x^4 - 8*x^5 - 48*x^6 - 56*x^7 + 116*x^9 +...
where the g.f. equals the infinite product:
A(x) = (1-x)^2/(1+x)^2 * (1-x^2)^4/(1+x^2)^4 * (1-x^3)^6/(1+x^3)^6 * (1-x^4)^8/(1+x^4)^8 * (1-x^5)^10/(1+x^5)^10 *...
The logarithm of the g.f. is illustrated by:
-log(A(x)) = 4*x + 16*x^2/2 + 40*x^3/3 + 64*x^4/4 + 104*x^5/5 + 160*x^6/6 + 200*x^7/7 + 256*x^8/8 +...+ 4*A076577(n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from Paul D. Hanna)

Cf. A156616, A076577, A001157 (sigma_2), A261386.

{a(n)=polcoeff(exp(sum(m=1, n+1, -(sigma(2*m,2)-sigma(m,2))*x^m/m+x*O(x^n))), n)}

{a(n)=polcoeff(prod(m=1,n,((1-x^m)/(1+x^m +x*O(x^n)))^(2*m)), n)}
for(n=0, 100, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} -(sigma_2(2*n) - sigma_2(n))*x^n/n ) where sigma_2(n) = sum of squares of divisors of n.

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

Original entry on oeis.org

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

Vladeta Jovovic, Oct 20 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A069733, A069184, A001157, A050999, A076577.

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 *)

a(n) = sumdiv(n, d, if ((d % 2) || (n/d % 2), d^2)); \\ Michel Marcus, Oct 30 2022

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

A374539 The sum of the squares of the infinitary divisors of n.

Original entry on oeis.org

1, 5, 10, 17, 26, 50, 50, 85, 82, 130, 122, 170, 170, 250, 260, 257, 290, 410, 362, 442, 500, 610, 530, 850, 626, 850, 820, 850, 842, 1300, 962, 1285, 1220, 1450, 1300, 1394, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2074, 2132, 2650, 2210, 2570, 2402, 3130, 2900

Offset: 1

Amiram Eldar, Jul 11 2024

Also the sum of the infinitary divisors of n^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A050376, A077609.

Similar sequences: A001157, A034676, A050999, A067558, A076577, A351265, A351307, A351647.

f[p_, e_] := p^(2^Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)]); a[1] = 1; a[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i = 1, #f~, b = binary(2 * f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)));}

from math import prod
from sympy import factorint
def A374539(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

a(n) = A049417(n^2).
a(n) <= A001157(n), with equality if and only if n is in A036537.
Multiplicative with a(p^e) = Product{k>=1, e_k=1} (p^(2^(k+1)) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Product_{P} (1 + 1/(P^2*(P+1))) = 1.14142906130350119631..., and P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).

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A216406 G.f.: Product_{n>=1} ((1-x^n)/(1+x^n))^(2*n).

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

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A374539 The sum of the squares of the infinitary divisors of n.

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