OFFSET
3,1
COMMENTS
Values are available for all 2 <= n <= 999 except for 12 and 24. If n is odd and > 3 then 2*n will have the same value in the sequence as n.
Additional terms starting with n = 12 are:
unknown, 53, 71, 331, 17, 239, 37, 213827, 1381, 673, 23, 47, unknown, 101, 53, 379, 29, 59, 331
The longest n*k+1 versus n*k-1 races up to n = 999 are for n = 3,6,8,12,24 and 168. When n = 168 the race ends at prime 273084304417.
The mod 12 and 24 races were checked by computer to 1.1 * 10^14 without n*k+1 ever leading.
Kevin Ford (private communication) provides the following information on these races: "My paper with Richard Hudson contains a lot of information about the location of sign changes for pi(x,q,a)-pi(x,q,b). Corollary 4 has rigorous upper bounds, but these will likely not be useful to you. The information in Tables 2 and 3 will be more helpful, as these provide the most likely places to look for the first sign change. In the case of the mod 12 race, it is probably around exp(187.536), or about 2.79 x 10^{81}. For the mod 24 race, it's about exp(43.453)=7.437... x 10^{18}".
In line with Ford and Hudson's prediction, for n=24, residue 1 leads -1 at 7390188907282602529; for n=12, residue 1 does not lead -1 up to 10^19. - Benjamin Chaffin, Jun 11 2026
REFERENCES
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.
Paulo Ribenboim, The Little Book of Big Primes, Springer 1991
LINKS
Kevin Ford and Richard H. Hudson, Sign changes in pi q,a(x) - pi q,b(x), Acta Arithmetica 100 (2001), 297-314.
Kevin Ford and Sergei Konyagin, Chebyshev's conjecture and the prime number race, arXiv:1910.08983 [math.NT], 2019.
Andrew Granville and Greg Martin, Prime Number Races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Andy Martin, Values for 2 to 999 as far as they are known
Andy Martin, C program
EXAMPLE
For n=5, the mod 5 values of the primes 2,3,5 and 7 are 2,3,0 and 2 respectively, so there is no change in the race. For the next prime 11, mod 5 gives 1, n*k+1 now leads 1 to 0, and the race is over.
PROG
(C) // See Links
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
Andy Martin, Aug 12 2016
EXTENSIONS
a(8)-a(11) from Andy Martin, Aug 15 2016
a(2) removed by Pontus von Brömssen, Jun 14 2026
STATUS
approved