A126830 Ramanujan numbers (A000594) read mod 729.

1, 705, 252, 715, 456, 513, 23, 645, 81, 720, 255, 117, 359, 177, 459, 70, 612, 243, 524, 177, 693, 441, 555, 702, 466, 132, 0, 407, 570, 648, 584, 495, 108, 621, 282, 324, 236, 546, 72, 333, 573, 135, 689, 75, 486, 531, 75, 144, 264, 480, 405, 77, 135, 0, 369, 255

Offset: 1

Author

N. J. A. Sloane, Feb 25 2007

Keywords

• nonn

References

• M. H. Ashworth, Congruence and identical properties of modular forms, Diss. University of Oxford, 1968.
• Oddmund Kolberg, Congruences for Ramanujan's Function ̈tau(n), Univ. Bergen Årbok Naturvit Rekke, No. 11, 1962.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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.

Crossrefs

Cf. A000594, A013959.

Programs

Mathematica

a[n_] := Mod[RamanujanTau[n], 729]; Array[a, 100] (* Amiram Eldar, Jan 05 2025 *)

PARI

a(n) = ramanujantau(n) % 729; \\ Amiram Eldar, Jan 05 2025

Formula

From Amiram Eldar, Jan 05 2025: (Start)

a(n) == 53 * sigma_11(n) (mod 729) for n == 2 (mod 3) (Kolberg, 1962).

a(n) == n^(-620) * sigma_1231(n) for n == 1 (mod 3) (Ashworth, 1968). (End)