This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A038691 #48 Jul 14 2025 14:49:28
%S A038691 1,3,7,13,89,2943,2945,2947,2949,2951,2953,50371,50375,50377,50379,
%T A038691 50381,50393,50413,50423,50425,50427,50429,50431,50433,50435,50437,
%U A038691 50439,50445,50449,50451,50503,50507,50515,50517,50821,50843,50853,50855,50857,50859,50861
%N A038691 Indices of primes at which the prime race 4k-1 vs. 4k+1 is tied.
%C A038691 Starting from a(27410) = 316064952537 the sequence includes the 8th sign-changing zone predicted by C. Bays et al back in 2001. The sequence with the first 8 sign-changing zones contains 419467 terms (see a-file) with a(419467) = 330797040309 as its last term. - _Sergei D. Shchebetov_, Oct 16 2017
%D A038691 Stan Wagon, The Power of Visualization, Front Range Press, 1994, pp. 2-3.
%H A038691 Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A038691/b038691.txt">Table of n, a(n) for n = 1..100000</a> (first 1000 terms from T. D. Noe)
%H A038691 A. Alahmadi, M. Planat and P. Solé, <a href="https://hal.archives-ouvertes.fr/hal-00650320">Chebyshev's bias and generalized Riemann hypothesis</a>, HAL Id: hal-00650320.
%H A038691 C. Bays and R. H. Hudson, <a href="http://dx.doi.org/10.1155/S0161171279000119">Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12</a>, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
%H A038691 C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, <a href="https://doi.org/10.1006/jnth.2000.2601">Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias</a>, J. Number Theory 87 (2001), pp. 54-76.
%H A038691 M. Deléglise, P. Dusart and X. Roblot, <a href="http://dx.doi.org/10.1090/S0025-5718-04-01649-7">Counting Primes in Residue Classes</a>, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 1565-1575.
%H A038691 A. Granville and G. Martin, <a href="https://web.archive.org/web/20240529054811/https://maa.org/sites/default/files/pdf/upload_library/22/Ford/granville1.pdf">Prime Number Races</a>, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
%H A038691 M. Rubinstein and P. Sarnak, <a href="https://projecteuclid.org/euclid.em/1048515870">Chebyshev’s bias</a>, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173-197.
%H A038691 Andrey S. Shchebetov and Sergei D. Shchebetov, <a href="/A038691/a038691-419467.zip">Table of n, a(n) for n = 1..419647 (zipped file)</a>
%H A038691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect.</a>
%e A038691 From _Jon E. Schoenfield_, Jul 24 2021: (Start)
%e A038691 a(n) is the n-th number m at which the prime race 4k-1 vs. 4k+1 is tied:
%e A038691 .
%e A038691 count
%e A038691 ----------
%e A038691 m p=prime(m) p mod 4 4k-1 4k+1
%e A038691 -- ---------- ------- ---- ----
%e A038691 1 2 2 0 = 0 a(1)=1
%e A038691 2 3 -1 1 0
%e A038691 3 5 +1 1 = 1 a(2)=3
%e A038691 4 7 -1 2 1
%e A038691 5 11 -1 3 1
%e A038691 6 13 +1 3 2
%e A038691 7 17 +1 3 = 3 a(3)=7
%e A038691 8 19 -1 4 3
%e A038691 9 23 -1 5 3
%e A038691 10 29 +1 5 4
%e A038691 11 31 -1 6 4
%e A038691 12 37 +1 6 5
%e A038691 13 41 +1 6 = 6 a(4)=13
%e A038691 (End)
%t A038691 Flatten[ Position[ FoldList[ Plus, 0, Mod[ Prime[ Range[ 2, 50900 ] ], 4 ]-2 ], 0 ] ]
%o A038691 (PARI) lista(nn) = {nbp = 0; nbm = 0; forprime(p=2, nn, if (((p-1) % 4) == 0, nbp++, if (((p+1) % 4) == 0, nbm++)); if (nbm == nbp, print1(primepi(p), ", ")););} \\ _Michel Marcus_, Nov 20 2016
%Y A038691 Cf. A002145, A002313, A007350, A007351, A038698, A051024, A051025, A066520, A096628, A096447, A096448, A199547.
%Y A038691 Cf. A156749; sequence showing Chebyshev bias in prime races (mod 4). - _Daniel Forgues_, Mar 26 2009
%K A038691 nonn
%O A038691 1,2
%A A038691 _Hans Havermann_