A134671 -id:A134671 - cp's OEIS frontend

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

Marc LeBrun

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

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

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).

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.

Cf. A000594, A013959, A121733, A121734, A098108, A126812-...

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

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

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 *)

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

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

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

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

A134670 First position k such that A046694(k) = A046694(k+1) =.. 0 are n consecutive zeros starting with A046694(k), where A046694 = Ramanujan tau numbers mod 691.

Original entry on oeis.org

1381, 16581, 290217, 1409635, 1118176194, 107792931954, 673751534392, 2587409974788

Offset: 1

Alexander Adamchuk, Nov 05 2007

Most probably a(5) = 1118176194, because it is a starting point of a string of 5 zeros, but the fact that this is the least such number needs to be confirmed.
Note that zeros of A046694(n) have the indices equal to the terms of arithmetic progressions of the type k*p, where primes p belong to A134671. Thus: a(1) = 1381 = 2*691 - 1, a(2) = 16581 = 3*5527 = 3*(8*691 - 1), a(3) = 290217 = 3*96739 = 3*(140*691 - 1), a(4) = 1409635 = 5*281927 = 5*(408*691 - 1), a(5) = 1118176194 = 6*186362699 = 6*(269700*691 - 1).
Also, note that all listed terms have the form a(n) = k*p - 1, where prime p is a prime of the form p = 2m*691 - 1 that belong to A134671. a(1) = 2*691 - 1, a(2) = 2*8291 - 1, a(3) = 2*145109 - 1, a(4) = 4*352409 - 1, a(5) = 5*223635239 - 1.

			a(1) = 1381 because A046694(1381) = 0 is the first zero in A046694(n).
a(2) = 16581 because A046694(16581) = A046694(16582) = 0 are the first two consecutive zeros in A046694(n).

Eric Weisstein's World of Mathematics, Ramanujan's Tau Function

Cf. A046694, A121733, A121734, A121742, A121743.

A134670 := proc(n)
    option remember;
    if n = 1 then
        1381 ;
    else
        for a from procname(n-1)+1 do
            wrks := true;
            for k from a to a+n-1 do
                if A046694(k) <> 0 then
                    wrks := false ;
                    break;
                end if;
            end do:
            if wrks then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Feb 01 2013

a(5) confirmed by Jud McCranie Oct 17 2020
a(6) by Jud McCranie Oct 17 2020
a(7) by Jud McCranie Oct 19 2020
a(8) by Jud McCranie Oct 22 2020

cp's OEIS Frontend

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

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