A046694 Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.

1, 667, 252, 601, 684, 171, 531, 178, 372, 168, 469, 123, 629, 385, 309, 388, 611, 55, 672, 630, 449, 491, 92, 632, 57, 106, 88, 580, 173, 185, 366, 666, 27, 538, 429, 379, 622, 456, 269, 136, 87, 280, 36, 632, 160, 556, 435, 345, 194, 14, 570, 52, 209, 652, 172, 542, 49

Offset: 1

Author

Marc LeBrun

Keywords

• easy
• nice
• nonn

Comments

Ramanujan tau is multiplicative, so this sequence is multiplicative mod 691.

There are pairs of identical terms a(n) and a(n+1). The first such twin pair is a(184) = a(185) = 483. The indices for a first twin in a pair are listed in A121733. Corresponding twin values are listed in A121734. - Alexander Adamchuk, Aug 18 2006

Set of values of a(n) consists of all integers from 0 to 690. The first a(n) = 0 occur at n = 2*691 - 1 = 1381 that is a prime. Set of numbers n such that a(n) = 0 is a union of all terms of the arithmetic progressions k*p, where p is a prime of the form p = 2m*691 - 1 and k>0 is an integer. Primes of the form p = 2m*691 - 1 are listed in A134671 = {1381,5527,8291,12437,22111,29021,30403,...}. It appears that in a(n) there are strings of consecutive zeros of any length. The first pair of consecutive zeros occurs at n = {16581,16582}. The least numbers k such that a(n) has a string of n consecutive zeros starting with a(k) are listed in A134670(n) = {1381,16581,290217,1409635,...}. - Alexander Adamchuk, Nov 05 2007

Examples

			Coefficient of x^2 in tau(x) = -24; 1^11+2^11 = 2049 = 667 mod 691 = -24 mod 691.
		

References

• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 169, (10.6.4).

Links

• T. D. Noe, 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, A121733, A121734, A098108, A126812-...

Cf. also A134670, A134671, A121742, A121743, A262339.

Programs

Maple

A046694 := proc(n)
    numtheory[sigma][11](n) mod 691 ;
end proc: # R. J. Mathar, Feb 01 2013

Mathematica

a[n_] := Mod[Total[Divisors[n]^11], 691]; a /@ Range[57] (* Jean-François Alcover , Apr 22 2011 *)
Table[Mod[DivisorSigma[11,n],691],{n,60}] (* Harvey P. Dale, Jun 01 2012 *)

PARI

a(n)=ramanujantau(n)%691 \\ Charles R Greathouse IV, Feb 08 2017

PARI

a(n)=sigma(n,11)%691 \\ Charles R Greathouse IV, Sep 09 2022

Python

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 11)%691 # Indranil Ghosh, Apr 24 2017

Formula

a(n) = A000594(n) mod A262339(6). - Jonathan Sondow, Sep 22 2015