A297406 -id:A297406 - cp's OEIS frontend

A007350 Where the prime race 4k-1 vs. 4k+1 changes leader.

3, 26861, 26879, 616841, 617039, 617269, 617471, 617521, 617587, 617689, 617723, 622813, 623387, 623401, 623851, 623933, 624031, 624097, 624191, 624241, 624259, 626929, 626963, 627353, 627391, 627449, 627511, 627733, 627919, 628013, 628427, 628937, 629371

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

The following references include some on the "prime race" question that are not necessarily related to this particular sequence. - N. J. A. Sloane, May 22 2006

Starting from a(12502) = A051025(27556) = 9103362505801, the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 194367 terms (see a-file) with a(194367) = 9543313015387 as its last term. - Sergei D. Shchebetov, Oct 13 2017

Ford, Kevin; Konyagin, Sergei; Chebyshev's conjecture and the prime number race. IV International Conference "Modern Problems of Number Theory and its Applications": Current Problems, Part II (Russian) (Tula, 2001), 67-91.
Granville, Andrew; Martin, Greg; Prime number races. (Spanish) With appendices by Giuliana Davidoff and Michael Guy. Gac. R. Soc. Mat. Esp. 8 (2005), no. 1, 197-240.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000 (terms 1..125 from Vincenzo Librandi, terms 126..301 from Robert G. Wilson v)
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.
Andrey Feuerverger and Greg Martin, Biases in the Shanks-Renyi prime number race Experiment. Math. 9 (2000), no. 4, 535-570.
Kevin Ford and Sergei Konyagin, Chebyshev's conjecture and the prime number race, 2002.
Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line, Duke Math. J., Volume 113, Number 2 (2002), 313-330.
Kevin Ford and Sergei Konyagin, The prime number race and zeros of L-functions off the critical line. II, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, 40 pp., Bonner Math. Schriften, 360, 2003.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Jerzy Kaczorowski, A contribution to the Shanks-Renyi race problem, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 176, 451-458.
Jerzy Kaczorowski, On the Shanks-Renyi race problem mod 5. J. Number Theory 50 (1995), no. 1, 106-118.
Greg Martin, Asymmetries in the Shanks-Renyi prime number race, arXiv:math/0010086 [math.NT], 2000; Number theory for the millennium, II (Urbana, IL, 2000), 403-415, A K Peters, Natick, MA, 2002.
J.-C. Puchta, On large oscillations of the remainder of the prime number theorems Acta Math. Hungar. 87 (2000), no. 3, 213-227.
M. Rubinstein and P. Sarnak, Chebyshev's bias, Exper. Math., 3 (1994), 173-197.
Andrey S. Shchebetov and Sergei D. Shchebetov, First 194367 terms (zipped file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
G. M. Ziegler, The great prime number record races, Notices Amer. Math. Soc. 51 (2004), no. 4, 414-416.

Cf. A007350, A007351, A007352, A007353, A007354, A297406, A297407, A297408, A297410, A297411, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A038698 (another way to watch this race).

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

lim = 10^5; k1 = 0; k3 = 0; t = Table[{p = Prime[k], If[Mod[p, 4] == 1, ++k1, k1], If[Mod[p, 4] == 3, ++k3, k3]}, {k, 2, lim}]; A007350 = {3}; Do[ If[t[[k-1, 2]] < t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] > t[[k+1, 3]] || t[[k-1, 2]] > t[[k-1, 3]] && t[[k, 2]] == t[[k, 3]] && t[[k+1, 2]] < t[[k+1, 3]], AppendTo[A007350, t[[k+1, 1]]]], {k, 2, Length[t]-1}]; A007350 (* Jean-François Alcover, Sep 07 2011 *)
lim = 10^5; k1 = 0; k3 = 0; p = 2; t = {}; parity = Mod[p, 4]; Do[p = NextPrime[p]; If[Mod[p, 4] == 1, k1++, k3++]; If[(k1 - k3)*(parity - Mod[p, 4]) > 0, AppendTo[t, p]; parity = Mod[p, 4]], {lim}]; t (* T. D. Noe, Sep 07 2011 *)

A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

Original entry on oeis.org

2, 608981813029, 608981813507, 608981813683, 608981813819, 608981814127, 608981814143, 608981818999, 608981820977, 608981826877, 608981826977, 608981827873, 608981828201, 608981836363, 608981836493, 608981836681, 608981836973, 608981836993, 608981837063

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, Apr 28 1994

nonn

Terms a(2n+1) form a subsequence of A098044.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence, although the terms are incorrect - see A185703).

Donovan Johnson, Table of n, a(n) for n = 1..39215
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2}(x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
G. Martin, Asymmetries in the Shanks-Renyi Prime number race, arXiv:math/0010086 [math.NT], 2000.
Pieter Moree, Chebyshev's bias for composite numbers with restricted prime divisors, Mathematics of computation 73.245 (2004): 425-449. See page 425.
M. Rubinstein and P. Sarnak, Chebyshev's Bias, Exper. Math. 3 (4) (1994) 209.
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A098044, A297406, A297407, A297408, A297410, A297411.

Terms from a(3) onwards corrected by Max Alekseyev, Feb 10 2011

A297408 Where the prime race among 10k+1, ..., 10k+9 changes leader.

Original entry on oeis.org

2, 13, 157, 193, 347, 383, 587, 673, 907, 1163, 1327, 1483, 1907, 1933, 2897, 4723, 5557, 5573, 6037, 6113, 6637, 6673, 7487, 8273, 8317, 8363, 8387, 8443, 8467, 8573, 8647, 8803, 8837, 8933, 9277, 9293, 10067, 10103, 11897, 11923, 12037, 12073, 12107, 12143

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

A007355 appears to be an erroneous version of this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A274121, A274122, A274123, A297406, A297407, A297408, A297410, A297411.

a297408(limit)={my(v=vector(10),vm=0,ivm=0,imv); forprime(p=2,limit,my(m=p%10);v[m]++;my(mv=vecmax(v,&imv));if(mv>vm,if(imv!=ivm,print1(p,", "); ivm=imv);vm=mv))};
a297408(12500) \\ Hugo Pfoertner, Jul 25 2021

from sympy import nextprime
from itertools import islice
def agen():
    c, p, leader = [0 for i in range(10)], 1, None
    while True:
        p = nextprime(p); last = p%10; c[last] += 1; m = max(c)
        if c.count(m) == 1 and c.index(m) == last and last != leader:
            yield p; leader = last
print(list(islice(agen(), 44))) # Michael S. Branicky, Dec 20 2022

A007353 Where the prime race among 5k+1, ..., 5k+4 changes leader.

Original entry on oeis.org

2, 83, 137, 293, 337, 443, 487, 523, 557, 743, 797, 1213, 1277, 1523, 1657, 1733, 1867, 1973, 2027, 2063, 2797, 2833, 2887, 4733, 5227, 5323, 5437, 5503, 5527, 5623, 5897, 5923, 6007, 6133, 6317, 6353, 6427, 6563, 6607, 6703, 7187, 7283, 7307, 7393, 7477, 8963

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

A007354 Where the prime race among 7k+1, ..., 7k+6 changes leader.

Original entry on oeis.org

2, 17, 131, 227, 733, 829, 929, 997, 1097, 1123, 1237, 1277, 1447, 1487, 1531, 1627, 1811, 1907, 1993, 2141, 2203, 2267, 2441, 2677, 2707, 3209, 3299, 3433, 3547, 3853, 4003, 4021, 4507, 4679, 4787, 4931, 5081, 5113, 7537, 7577, 7649, 7759, 7817, 8039, 8461, 8543

Offset: 1

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

primerace(mod,limit)={my(v=vector(mod), vm=0, ivm=0, imv); forprime(p=2, limit, my(m=p%mod); v[m+1]++;  v[m+1]++; my(mv=vecmax(v, &imv)); if(mv>vm, if(imv!=ivm, print1(p, ", "); ivm=imv); vm=mv))};
primerace(7,8600) \\ Hugo Pfoertner, Jul 25 2021

A297407 Where the prime race among 9k+1, ..., 9k+8 changes leader.

Original entry on oeis.org

2, 167, 191, 419, 461, 563, 587, 617, 677, 761, 857, 881, 929, 1427, 1451, 1607, 1667, 1777, 1823, 1867, 1913, 2351, 2399, 2459, 4127, 4583, 5039, 5087, 5171, 7283, 7349, 7517, 7547, 7643, 7691, 7901, 8681, 8837, 8933, 11243, 11903, 11927, 18329, 18371, 19913

Offset: 1

Sean A. Irvine, Dec 29 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993

Cf. A007352, A007350, A007353, A007354, A297406, A297407, A297408, A297410, A297411.

V:= <(0$8)>:
lead:= 1:
Res:= NULL:
count:= 0:
p:= 1:
while count < 100 do
  p:= nextprime(p);
  t:= p mod 9;
  V[t]:= V[t]+1;
  if V[t] > V[lead] then
    lead:= t;
    Res:= Res, p;
    count:= count+1;
  fi
od:
Res; # Robert Israel, Jan 01 2018

cp's OEIS Frontend

A007350 Where the prime race 4k-1 vs. 4k+1 changes leader.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

A007352 Where the prime race 3k-1 vs. 3k+1 changes leader.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A297408 Where the prime race among 10k+1, ..., 10k+9 changes leader.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Python

A007353 Where the prime race among 5k+1, ..., 5k+4 changes leader.

Views

Author

Keywords

References

Links

Crossrefs

A007354 Where the prime race among 7k+1, ..., 7k+6 changes leader.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

A297407 Where the prime race among 9k+1, ..., 9k+8 changes leader.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A297410 Where the prime race among 11k+1, ..., 11k+10 changes leader.

Views

Author

Keywords

Links

Crossrefs

A297411 Where the prime race among 12k+1, ..., 12k+11 changes leader.

Views

Author

Keywords

Links

Crossrefs