A045823 a(n) = sigma_3(2*n+1).
1, 28, 126, 344, 757, 1332, 2198, 3528, 4914, 6860, 9632, 12168, 15751, 20440, 24390, 29792, 37296, 43344, 50654, 61544, 68922, 79508, 95382, 103824, 117993, 137592, 148878, 167832, 192080, 205380, 226982, 260408, 276948, 300764, 340704, 357912
Offset: 0
Examples
q + 28*q^3 + 126*q^5 + 344*q^7 + 757*q^9 + 1332*q^11 + 2198*q^13 + ...
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
Programs
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Magma
[DivisorSigma(3, 2*n+1): n in [0..40]]; // Vincenzo Librandi, Jun 02 2019
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Maple
A045823 := proc(n) numtheory[sigma][3](2*n+1) ; end proc: seq(A045823(n),n=0..30) ; # R. J. Mathar, Nov 25 2018
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Mathematica
DivisorSigma[3, Range[1, 75, 2]] (* Harvey P. Dale, Jan 11 2015 *)
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PARI
{a(n) = if( n<0, 0, sigma(2 * n + 1, 3))} /* Michael Somos, Nov 29 2007 */ -
PARI
{a(n) = local(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( (eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8) / (2 * eta(x + A)^8 * eta(x^2 + A)^8), n))} /* Michael Somos, Nov 29 2007 */
Formula
Expansion of q^(-1) * ( E_4(q) - 9 * E_4(q^2) + 8 * E_4(q^4) ) / 240 in powers of q^2. - Michael Somos, Nov 29 2007
Expansion of q^(-1) * (eta(q^2)^24 + eta(q)^16 * eta(q^4)^8) / (2 * eta(q)^8 * eta(q^2)^8) in powers of q^2. - Michael Somos, Nov 29 2007
a(n) = b(2*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1) if p>2. - Michael Somos, Nov 29 2007
G.f.: (theta_3(q)^8 - theta_4(q)^8) / (32*q) = Sum_{n>=0} sigma_3(2*n+1)*q^(2*n). - Paul D. Hanna, Jun 02 2018
Sum_{k=0..n} a(k) ~ (15*zeta(4)/8) * n^4. - Amiram Eldar, Dec 12 2023
Extensions
More terms from Benoit Cloitre, Apr 12 2003