A013955 -id:A013955 - cp's OEIS frontend

A126827 Ramanujan numbers (A000594) read mod 27.

1, 3, 9, 13, 24, 0, 23, 24, 0, 18, 12, 9, 8, 15, 0, 16, 18, 0, 11, 15, 18, 9, 15, 0, 7, 24, 0, 2, 3, 0, 17, 9, 0, 0, 12, 0, 20, 6, 18, 9, 6, 0, 14, 21, 0, 18, 21, 9, 21, 21, 0, 23, 0, 0, 18, 12, 18, 9, 24, 0, 23, 24, 0, 10, 3, 0, 8, 18, 0, 9, 18, 0, 11, 6, 9, 8, 6, 0, 5, 6, 0, 18, 3, 18, 0, 15, 0

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
George E. Andrews and Bruce C. Berndt, Ramanujan's Unpublished Manuscript on the Partition and Tau Functions, in: Ramanujan's Lost Notebook, Part III, Springer, New York, NY, 2012.
R. P. Bambah and S. Chowla, Congruence properties of Ramanujan’s function tau(n), Bull. Amer. Math. Soc., Vol. 53 (1947), pp. 950-955.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A013955.

a[n_] := Mod[RamanujanTau[n], 27]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

a(n) = ramanujantau(n) % 27; \\ Amiram Eldar, Jan 05 2025

a(n) == n^2 * sigma_7(n) (mod 27) (Bambah and Chowla, 1947, eq. (16), p. 954; Andrews and Berndt, 2012, eq. (5.12.3), p. 114). - Amiram Eldar, Jan 05 2025

A126828 Ramanujan numbers (A000594) read mod 81.

Original entry on oeis.org

1, 57, 9, 67, 51, 27, 23, 78, 0, 72, 12, 36, 35, 15, 54, 70, 45, 0, 38, 15, 45, 36, 69, 54, 61, 51, 0, 2, 3, 0, 17, 9, 27, 54, 39, 0, 74, 60, 72, 9, 6, 54, 41, 75, 0, 45, 75, 63, 21, 75, 0, 77, 54, 0, 45, 12, 18, 9, 24, 54, 50, 78, 0, 37, 3, 0, 62, 18, 54, 36, 18, 0, 65, 6, 63, 35, 33, 54

Offset: 1

N. J. A. Sloane, Feb 25 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. P. Bambah and S. Chowla, Congruence properties of Ramanujan’s function tau(n), Bull. Amer. Math. Soc. 53 (1947), 950-955.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.

Cf. A000594, A013955.

Mod[RamanujanTau@ #, 81] & /@ Range@ 78 (* Michael De Vlieger, Apr 26 2016 *)

a(n) = ramanujantau(n) % 81; \\ Amiram Eldar, Jan 05 2025

a(n) = (n^2+k)*sigma_7(n) with k=0 if gcd(n,3)=1 and k=9 if gcd(n,3)=2. - Michel Marcus, Apr 26 2016

A029817 Average theta series of odd unimodular lattices of dimension 16 (multiplied by 17).

Original entry on oeis.org

17, 32, 4064, 70016, 528352, 2500032, 8892032, 26353408, 67637216, 153125024, 317504064, 623589504, 1156034176, 2007952576, 3346882816, 5470070016, 8657571808, 13130837568, 19446878048, 28603895680, 41278028352, 57661256704, 79195867008, 108954414336, 147990228608

Offset: 0

N. J. A. Sloane

nonn

Heng Huat Chan and Christian Krattenthaler, Recent progress in the study of representations of integers as sums of squares, Bulletin of the London Mathematical Society, Vol. 37, No. 6 (2005), pp. 818-826; arXiv preprint, arXiv:math/0407061 [math.NT], 2004.

Cf. A013955, A321811.

max = 20; s = 17 + 32*Sum[k^7*q^k/(1-(-q)^k), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Dec 07 2015 *)

a(n)=if(n<1,17*(n==0),32*sumdiv(n,d,d^7-2*if(d%4==2,(d/2)^7))) /* Michael Somos, Jul 16 2004 */

G.f.: 17 + 32 * Sum_{k >= 1} k^7*q^k/(1-(-q)^k).

a(n) = 32 * (-1)^n * (A013955(n) - 2 * A321811(2*n)) for n >= 1. - Amiram Eldar, Jan 07 2025

cp's OEIS Frontend

A126827 Ramanujan numbers (A000594) read mod 27.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A126828 Ramanujan numbers (A000594) read mod 81.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A029817 Average theta series of odd unimodular lattices of dimension 16 (multiplied by 17).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula