A121742 -id:A121742 - 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

A121743 Values of the Ramanujan tau triples mod 691 such that three consecutive Ramanujan tau numbers are congruent mod 691.

Original entry on oeis.org

0, 276, 91, 79, 0, 0, 0, 0, 76, 349, 212, 355, 662, 227, 342, 616, 182, 641, 105, 0, 21, 33, 0, 0, 316, 436, 346, 109, 468, 557, 261, 512, 299, 532, 565, 214, 72, 218, 436, 0, 166, 532, 0, 591, 0, 144, 0, 544, 257, 0, 0, 0, 422, 0, 0, 488, 0, 0, 0, 488, 0, 233, 371, 0, 380, 28, 0, 641, 414, 331, 0, 487, 0, 666, 130, 14, 0, 0, 321, 620, 0, 339, 533

Offset: 1

Alexander Adamchuk, Aug 19 2006

nonn

Corresponding indices of the Ramanujan tau triples mod 691 are listed in A121742. All a(n) belong to the Ramanujan tau twins mod 691 A121734(n). There are also quadruplets in the Ramanujan tau mod 691 such that A046694(n) = A046694(n+1) = A046694(n+2) = A046694(n+3). The first such Ramanujan tau quadruplet mod 691 starts with A046694(1409635) = 0.

Jud McCranie, Table of n, a(n) for n = 1..2568
Eric Weisstein Ramanujan's Tau Function.

Cf. A000594, A046694, A121733, A121734, A121742.

Do[f=Mod[DivisorSigma[11,n],691];g=Mod[DivisorSigma[11,n+1],691];h=Mod[DivisorSigma[11,n+2],691];If[f==g&&g==h,Print[{n,f}]],{n,1,1500000}]
Select[Partition[Table[Mod[DivisorSigma[11,n],691],{n,10000000}],3,1],Length[ Union[#]]==1&][[All,1]] (* Harvey P. Dale, Jan 31 2020 *)

a(n) = A000594(A121742(n)) mod 691.

a(n) = A046694(A121742(n)).

a(7)-a(16) from Amiram Eldar, Jan 26 2020

More terms by Jud McCranie Nov 02 2020

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

A134671 Primes of the form 2m*691 - 1.

Original entry on oeis.org

1381, 5527, 8291, 12437, 22111, 29021, 30403, 34549, 37313, 42841, 51133, 53897, 58043, 62189, 70481, 92593, 96739, 105031, 120233, 134053, 145109, 167221, 179659, 182423, 186569, 187951, 192097, 194861, 212827, 216973, 233557, 281927

Offset: 1

Alexander Adamchuk, Nov 05 2007

Note that all zeros of A046694(n) have the indices equal to the terms of all arithmetic progressions of the type k*p, where primes p belong to a(n). Thus A046694(k*a(n)) = 0 for all integer k > 0.

			a(1) = 1381 = 2*691 - 1 is a first prime of the form 2m*691 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ramanujan's Tau Function.

Cf. A046694 = Ramanujan tau numbers mod 691 = sum of 11th power of divisors mod 691.
Cf. A121733 = Numbers n such that two consecutive Ramanujan tau numbers are congruent mod 691.
Cf. A121734 = Ramanujan tau numbers such that A000594[n] == A000594[n+1] mod 691.
Cf. A121742 = Numbers n such that three consecutive Ramanujan tau numbers are congruent mod 691.
Cf. A121743 = Values of the Ramanujan tau triples mod 691 such that three consecutive Ramanujan tau numbers are congruent mod 691.
Cf. A134670 = Least number k such that A046694 has a string of n consecutive zeros starting with A046694(k).

[a: n in [0..250] | IsPrime(a) where a is 1382*n-1]; // Vincenzo Librandi, Nov 07 2014

Select[ 2*691*Range[ 1000 ] - 1, PrimeQ[ # ] & ]
Select[Table[1382 n - 1, {n, 0, 300}], PrimeQ] (* Vincenzo Librandi, Nov 07 2014 *)

list(lim)=my(v=List()); forprimestep(p=1381,lim,Mod(-1,1382), listput(v,p)); Vec(v) \\ Charles R Greathouse IV, Sep 09 2022

A337916 Numbers k such that four (or more) consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) == A000594(k+3) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2) = A046694(k+3).

Original entry on oeis.org

1409635, 74627996, 98011434, 109454388, 153236152, 191545190, 392460356, 427203833, 497768756, 504982791, 538208841, 538565394, 541301742, 549096232, 590970836, 591800035, 623392556, 660872395, 698269314, 779613836, 796944116, 839316232, 884093036, 1045217422, 1118176194

Offset: 1

Jud McCranie, Nov 02 2020

nonn

The corresponding Ramanujan tau numbers mod 691 are listed in A337917(n) = A046694(a(n)). See A121742 for more information.

			Ramajuan tau A000594(1045217422) == A000594(1045217423) == A000594(1045217424) == A000594(1045217425) == 269, so 1045217422 is in the sequence.

Jud McCranie, Table of n, a(n) for n = 1..1419
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A121742, A337917, A000594, A046694, A121733, A121734, A121743.

A337917 Values of the Ramanujan tau quadruples mod 691 such that four (or more) consecutive Ramanujan tau numbers are congruent mod 691.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 269, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 253, 0, 629, 0, 0, 0, 0, 0, 0, 200, 360, 0, 0, 0, 484, 0, 0, 0, 0, 0, 0, 0, 584, 0, 0, 512, 464, 417, 0, 0, 0, 0, 0, 0, 0, 367, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 466, 161, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 463, 0, 0

Offset: 1

Jud McCranie, Nov 02 2020

nonn

Corresponding indices of the Ramanujan tau quadruples mod 691 are listed in A337916.

tau(n) mod 691 appears to be almost evenly distributed between 0 and 690 - 2 through 690 each occur approximately 1/691 of the time; 0 occurs slightly more than that and 1 occurs slightly less. However, consecutive values that are congruent mod 691 are predominately congruent to 0.

			Ramajuan tau A000594(1045217422) == A000594(1045217423) == A000594(1045217424) == A000594(1045217425) == 269
so 269 is in the sequence, corresponding to 1045217422 in A337916.

Jud McCranie, Table of n, a(n) for n = 1..1419

Cf. A337916, A121742, A121743, A000594, A046694, A121733, A121734, A121742.

cp's OEIS Frontend

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

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A121743 Values of the Ramanujan tau triples mod 691 such that three consecutive Ramanujan tau numbers are congruent mod 691.

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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.

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A134671 Primes of the form 2m*691 - 1.

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A337916 Numbers k such that four (or more) consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) == A000594(k+3) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2) = A046694(k+3).

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A337917 Values of the Ramanujan tau quadruples mod 691 such that four (or more) consecutive Ramanujan tau numbers are congruent mod 691.

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