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

%I A099978 #26 Jun 28 2025 01:10:46
%S A099978 1,10,26,50,91,122,170,260,290,362,500,530,651,820,842,962,1220,1300,
%T A099978 1370,1700,1682,1850,2366,2210,2451,2900,2810,3172,3620,3482,3722,
%U A099978 4550,4420,4490,5300,5042,5330,6510,6100,6242,7381,6890,7540,8420,7922,8500
%N A099978 Bisection of A001157: a(n) = sigma_2(2n-1).
%H A099978 Amiram Eldar, <a href="/A099978/b099978.txt">Table of n, a(n) for n = 1..10000</a>
%F A099978 a(n) = A001157(2n-1) = sigma_2(2n-1). - _M. F. Hasler_, Mar 06 2017
%F A099978 Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/6. - _Vaclav Kotesovec_, Aug 07 2022
%F A099978 G.f.: Sum_{n>=1} (2*n-1)^2 * x^n / (1 - x^(2*n-1)). - _Paul D. Hanna_, Jun 23 2025
%e A099978 From _M. F. Hasler_, Mar 06 2017: (Start)
%e A099978 a(1) = sigma_2(2*1-1) = 1.
%e A099978 a(2) = sigma_2(2*2-1) = 1 + 3^2 = 10.
%e A099978 a(5) = sigma_2(2*5-1) = 1 + 3^2 + 9^2 = 91. (End)
%e A099978 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
%p A099978 with(numtheory): A099978 := n->sigma[2](2*n-1): seq(A099978(n), n=1..60);
%t A099978 DivisorSigma[2, Range[1, 91, 2]] (* _Amiram Eldar_, Aug 17 2019 *)
%o A099978 (PARI) A099978(n)=sigma(2*n-1,2) \\ _M. F. Hasler_, Mar 06 2017
%Y A099978 Cf. A099979(n) = sigma_2(2n), the other bisection of A001157.
%Y A099978 Cf. A283224.
%K A099978 nonn,easy
%O A099978 1,2
%A A099978 _N. J. A. Sloane_, Nov 19 2004
%E A099978 More terms from _Emeric Deutsch_, Dec 07 2004
%E A099978 Edited by _M. F. Hasler_, Mar 06 2017