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%I #23 Sep 08 2022 08:45:09
%S 1,13,70,247,671,1547,3178,5916,10317,17088,26818,40703,60034,85463,
%T 119288,163736,218924,288933,377482,482734,612535,772291,955604,
%U 1177050,1443522,1742481,2097702,2517368,2978851,3519151,4152486,4836104,5625521,6543616,7517622
%N a(n) = Sum_{k=0..n-1} sigma(2k+1)*sigma_3(n-k).
%C An amazing Ramanujan identity. Here sigma_m(n) denotes Sum_{d|n} d^m.
%D Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; page 301.
%H Robert Israel, <a href="/A081860/b081860.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = (1/240)*(sigma_5(2n+1)-sigma(2n+1)) (see A081863(2)).
%p f:= n -> 1/240*(numtheory:-sigma[5](2*n+1)-numtheory:-sigma(2*n+1)):
%p map(f, [$1..100]); # _Robert Israel_, Aug 12 2018
%t lst={}; Do[AppendTo[lst, DivisorSigma[5, 2 n + 1] - DivisorSigma[1, 2 n + 1]], {n, 40}]; lst / 240 (* _Vincenzo Librandi_, Aug 13 2018 *)
%t Table[Sum[DivisorSigma[1,2k+1]DivisorSigma[3,n-k],{k,0,n-1}],{n,35}] (* _Harvey P. Dale_, Jul 25 2020 *)
%o (PARI) a(n) = sum(k=0, n-1, sigma(2*k+1)*sigma(n-k, 3)); \\ _Michel Marcus_, Dec 04 2013
%o (PARI) a(n) = (sigma(2*n+1, 5) - sigma(2*n+1))/240; \\ _Michel Marcus_, Dec 04 2013
%o (Magma) [(DivisorSigma(5, 2*n+1)-DivisorSigma(1, 2*n+1))/240: n in [1..40]]; // _Vincenzo Librandi_, Aug 13 2018
%Y Cf. A000203, A001158, A001160.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Apr 11 2003
%E Three more terms from _Michel Marcus_, Dec 04 2013
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