|
%I #15 Dec 12 2023 08:00:54
%S 1,73,344,1134,2198,4681,6860,11988,15751,25112,29792,44226,50654,
%T 73710,79508,109512,117993,160454,167832,219510,226982,299593,300764,
%U 390096,389018,500780,493040,620298,619164,779220,756112,934416,912674,1149823,1092728
%N a(n) = sigma_3(3n+1).
%F Expansion of q^(-1/3) * c(q) * (c(q)^3 + b(q)^3 / 3) in powers of q where b(), c() are cubic AGM functions. - _Michael Somos_, Aug 22 2007
%F If b(3*n) = 0, b(3*n+1) = a(n), b(3*n+2) = A092343(n), then b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) otherwise. - _Michael Somos_, Aug 22 2007
%F a(n) = A000731(n) + 81*A033690(n-1). - _Michael Somos_, Aug 22 2007
%F Sum_{k=0..n} a(k) ~ (20*zeta(4)/3) * n^4. - _Amiram Eldar_, Dec 12 2023
%e q + 73*q^4 + 344*q^7 + 1134*q^10 + 2198*q^13 + 4681*q^16 + ...
%t DivisorSigma[3,3*Range[0,40]+1] (* _Harvey P. Dale_, Apr 22 2019 *)
%o (PARI) {a(n) = if(n<0, 0, sigma(3*n+1, 3))} /* _Michael Somos_, Aug 22 2007 */
%Y Trisection of A001158.
%Y Cf. A000731, A013662, A033690, A045823, A091986, A092341, A092343.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Mar 20 2004
|
