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A038691 Indices of primes at which the prime race 4k-1 vs. 4k+1 is tied. 23
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 (list; graph; refs; listen; history; text; internal format)
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. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - Daniel Forgues, Mar 26 2009
Sequence in context: A137474 A071087 A309775 * A237890 A082718 A221211
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 18 11:53 EDT 2024. Contains 372630 sequences. (Running on oeis4.)