A038691 Indices of primes at which the prime race 4k-1 vs. 4k+1 is tied.

1, 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, 50375, 50377, 50379, 50381, 50393, 50413, 50423, 50425, 50427, 50429, 50431, 50433, 50435, 50437, 50439, 50445, 50449, 50451, 50503, 50507, 50515, 50517, 50821, 50843, 50853

Offset

1,2

Comments

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

References

Stan Wagon, The Power of Visualization, Front Range Press, 1994, pp. 2-3.

Links

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (first 1000 terms from T. D. Noe)

A. Alahmadi, M. Planat and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.

C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.

C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp. 54-76.

M. Deléglise, P. Dusart and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 1565-1575.

A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.

M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173-197.

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..419647 (zipped file)

Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Example

From Jon E. Schoenfield, Jul 24 2021: (Start)
a(n) is the n-th number m at which the prime race 4k-1 vs. 4k+1 is tied:
.
                             count
                           ----------
   m  p=prime(m)  p mod 4  4k-1  4k+1
  --  ----------  -------  ----  ----
   1       2         2       0  =  0    a(1)=1
   2       3        -1       1     0
   3       5        +1       1  =  1    a(2)=3
   4       7        -1       2     1
   5      11        -1       3     1
   6      13        +1       3     2
   7      17        +1       3  =  3    a(3)=7
   8      19        -1       4     3
   9      23        -1       5     3
  10      29        +1       5     4
  11      31        -1       6     4
  12      37        +1       6     5
  13      41        +1       6  =  6    a(4)=13
(End)

Mathematica

Flatten[ Position[ FoldList[ Plus, 0, Mod[ Prime[ Range[ 2, 50900 ] ], 4 ]-2 ], 0 ] ]

Prog

(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

Crossrefs

Cf. A002145, A002313, A007350, A007351, A038698, A051024, A051025, A066520, A096628, A096447, A096448, A199547

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - Daniel Forgues, Mar 26 2009

Sequence in context: A137474 A071087 A309775 * A237890 A082718 A221211

Adjacent sequences: A038688 A038689 A038690 * A038692 A038693 A038694

Keyword

nonn

Author

Hans Havermann

Status

approved