A081860 a(n) = Sum_{k=0..n-1} sigma(2k+1)*sigma_3(n-k).

1, 13, 70, 247, 671, 1547, 3178, 5916, 10317, 17088, 26818, 40703, 60034, 85463, 119288, 163736, 218924, 288933, 377482, 482734, 612535, 772291, 955604, 1177050, 1443522, 1742481, 2097702, 2517368, 2978851, 3519151, 4152486, 4836104, 5625521, 6543616, 7517622

Offset

1,2

Comments

An amazing Ramanujan identity. Here sigma_m(n) denotes Sum_{d|n} d^m.

References

Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; page 301.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

a(n) = (1/240)*(sigma_5(2n+1)-sigma(2n+1)) (see A081863(2)).

Maple

f:= n -> 1/240*(numtheory:-sigma[5](2*n+1)-numtheory:-sigma(2*n+1)):
map(f, [$1..100]); # Robert Israel, Aug 12 2018

Mathematica

lst={}; Do[AppendTo[lst, DivisorSigma[5, 2 n + 1] - DivisorSigma[1, 2 n + 1]], {n, 40}]; lst  / 240 (* Vincenzo Librandi, Aug 13 2018 *)
Table[Sum[DivisorSigma[1, 2k+1]DivisorSigma[3, n-k], {k, 0, n-1}], {n, 35}] (* Harvey P. Dale, Jul 25 2020 *)

Prog

(PARI) a(n) = sum(k=0, n-1, sigma(2*k+1)*sigma(n-k, 3)); \\ Michel Marcus, Dec 04 2013
(PARI) a(n) = (sigma(2*n+1, 5) - sigma(2*n+1))/240; \\ Michel Marcus, Dec 04 2013
(Magma) [(DivisorSigma(5, 2*n+1)-DivisorSigma(1, 2*n+1))/240: n in [1..40]]; // Vincenzo Librandi, Aug 13 2018

Crossrefs

Cf. A000203, A001158, A001160.

Sequence in context: A146469 A146381 A085461 * A050403 A235454 A296831

Adjacent sequences: A081857 A081858 A081859 * A081861 A081862 A081863

Keyword

nonn

Author

Benoit Cloitre, Apr 11 2003

Extensions

Three more terms from Michel Marcus, Dec 04 2013

Status

approved