cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

Benoit Cloitre, Apr 11 2003

Keywords

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

Crossrefs

Cf. A000203, A001158, A001160.

Programs

  • Magma
    [(DivisorSigma(5, 2*n+1)-DivisorSigma(1, 2*n+1))/240: n in [1..40]]; // Vincenzo Librandi, Aug 13 2018
  • 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 *)
  • 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

Formula

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

Extensions

Three more terms from Michel Marcus, Dec 04 2013

A081861 a(n) = (1/24)*(sigma_3(2*n-1) - sigma_1(2*n-1)).

Original entry on oeis.org

0, 1, 5, 14, 31, 55, 91, 146, 204, 285, 400, 506, 655, 850, 1015, 1240, 1552, 1804, 2109, 2562, 2870, 3311, 3971, 4324, 4914, 5730, 6201, 6990, 8000, 8555, 9455, 10846, 11536, 12529, 14192, 14910, 16206, 18371, 19088, 20540, 22990, 23821, 25794
Offset: 1

Views

Author

Benoit Cloitre, Apr 11 2003

Keywords

Links

Crossrefs

Cf. A000203, A001158, A008438, A045823, A081863(1).

Programs

  • Mathematica
    Table[(DivisorSigma[3,2n-1]-DivisorSigma[1,2n-1])/24,{n,50}] (* Harvey P. Dale, Dec 15 2012 *)
  • PARI
    a(n) = my(f = factor(2*n-1)); (sigma(f, 3) - sigma(f)) / 24; \\ Amiram Eldar, Jan 04 2025

Formula

a(n) = (A045823(n-1) - A008438(n-1)) / 24. - Amiram Eldar, Jan 04 2025

A081862 (1/168)*(sigma_7(2n-1)-sigma_1(2n-1)).

Original entry on oeis.org

0, 13, 465, 4902, 28483, 115995, 373503, 1017498, 2442492, 5320665, 10725680, 20266818, 36330915, 62292490, 102677835, 163765560, 253797216, 382977372, 565070697, 817224746, 1159251630, 1617967923, 2225268903, 3015613812
Offset: 1

Views

Author

Benoit Cloitre, Apr 11 2003

Keywords

Crossrefs

Cf. A000203, A081863(3).

Programs

  • Mathematica
    Table[(DivisorSigma[7,n]-DivisorSigma[1,n])/168,{n,1,49,2}] (* Harvey P. Dale, Sep 17 2013 *)
Showing 1-3 of 3 results.

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