OFFSET
1,2
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
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.
FORMULA
EXAMPLE
Coefficient of x^2 in tau(x) = -24; 1^11+2^11 = 2049 = 667 mod 691 = -24 mod 691.
MAPLE
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
STATUS
approved