A099978 Bisection of A001157: a(n) = sigma_2(2n-1).

1, 10, 26, 50, 91, 122, 170, 260, 290, 362, 500, 530, 651, 820, 842, 962, 1220, 1300, 1370, 1700, 1682, 1850, 2366, 2210, 2451, 2900, 2810, 3172, 3620, 3482, 3722, 4550, 4420, 4490, 5300, 5042, 5330, 6510, 6100, 6242, 7381, 6890, 7540, 8420, 7922, 8500

Offset: 1

Author

N. J. A. Sloane, Nov 19 2004

Keywords

• nonn
• easy

Examples

			From _M. F. Hasler_, Mar 06 2017: (Start)
a(1) = sigma_2(2*1-1) = 1.
a(2) = sigma_2(2*2-1) = 1 + 3^2 = 10.
a(5) = sigma_2(2*5-1) = 1 + 3^2 + 9^2 = 91. (End)
G.f.: A(x) = x + 10*x^2 + 26*x^3 + 50*x^4 + 91*x^5 + 122*x^6 + 170*x^7 + 260*x^8 + 290*x^9 + 362*x^10 + 500*x^11 + 530*x^12 + ... where A(x) = x/(1 - x) + 3^2*x^2/(1 - x^3) + 5^2*x^3/(1 - x^5) + 7^2*x^4/(1 - x^7) + 9^2*x^5/(1 - x^9) + .... - _Paul D. Hanna_, Jun 23 2025
		

Links

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

Crossrefs

Cf. A099979(n) = sigma_2(2n), the other bisection of A001157.

Cf. A283224.

Programs

Maple

with(numtheory): A099978 := n->sigma[2](2*n-1): seq(A099978(n), n=1..60);

Mathematica

DivisorSigma[2, Range[1, 91, 2]] (* Amiram Eldar, Aug 17 2019 *)

PARI

A099978(n)=sigma(2*n-1,2) \\ M. F. Hasler, Mar 06 2017

Formula

a(n) = A001157(2n-1) = sigma_2(2n-1). - M. F. Hasler, Mar 06 2017

Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/6. - Vaclav Kotesovec, Aug 07 2022

G.f.: Sum_{n>=1} (2*n-1)^2 * x^n / (1 - x^(2*n-1)). - Paul D. Hanna, Jun 23 2025

Extensions

More terms from Emeric Deutsch, Dec 07 2004

Edited by M. F. Hasler, Mar 06 2017