

A038691


Prime race 4k1 vs. 4k+1 is tied at nth prime.


21



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
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OFFSET

1,2


COMMENTS

Starting from a(27410) = 316064952537 the sequence includes the 8th signchanging zone predicted by C. Bays et al back in 2001. The sequence with the first 8 signchanging zones contains 419467 terms (see afile) 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. 23.


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, P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal00650320.
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. 111119, 1979.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet Lfunctions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp. 5476.
M. Deléglise, P. Dusart, X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 15651575.
A. Granville, G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 133.
M. Rubinstein, P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173197.
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.


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 (((p1) % 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



