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A121742 Numbers k such that three consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2).

%I A121742 #24 Feb 16 2025 08:33:02
%S A121742 290217,477155,1051085,1153412,1409635,1409636,1641812,2056412,
%T A121742 2657865,2945116,3724928,4570784,5115359,5187777,5567783,5720418,
%U A121742 7836078,8736807,8932428,9618716,9957630,10175867,10447914,10547421,10982172,11359120,11499876,11735611,12651355,13018169,13515452,13867914
%N A121742 Numbers k such that three consecutive Ramanujan tau numbers are congruent mod 691, or A000594(k) == A000594(k+1) == A000594(k+2) mod 691, or A046694(k) = A046694(k+1) = A046694(k+2).
%C A121742 Corresponding Ramanujan tau numbers mod 691 are listed in A121743(n) = A046694(a(n)). A121743(n) begins {0,276,91,79,0,0,...}. a(n) are the indices of the first number in the Ramanujan tau triples mod 691. All a(n) belong to A121733(n) - indices of the first number in the Ramanujan tau twins mod 691. 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.
%H A121742 Jud McCranie, <a href="/A121742/b121742.txt">Table of n, a(n) for n = 1..2568</a>
%H A121742 Eric Weisstein <a href="https://mathworld.wolfram.com/TauFunction.html">Ramanujan's Tau Function</a>.
%t 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}]
%Y A121742 Cf. A000594, A046694, A121733, A121734, A121743.
%K A121742 nonn
%O A121742 1,1
%A A121742 _Alexander Adamchuk_, Aug 19 2006
%E A121742 a(7)-a(16) from _Amiram Eldar_, Jan 26 2020
%E A121742 More terms from _Jud McCranie_, Nov 02 2020