A055626 First prime starting a chain of exactly n consecutive primes congruent to 5 modulo 6.

5, 23, 47, 251, 1889, 7793, 43451, 243161, 726893, 759821, 2280857, 1820111, 10141499, 40727657, 19725473, 136209239, 744771077, 400414121, 1057859471, 489144599, 13160911739, 766319189, 38451670931, 119618704427, 21549657539, 141116164769, 140432294381, 437339303279

Offset: 1

Author

Labos Elemer, Jun 05 2000

Keywords

• nonn

Comments

The term "exactly" means that before the first and after the last primes of chain, the immediate primes are not congruent to 5 modulo 6.

a(21)>2^31, a(22)= 766319189. - Hugo Pfoertner, Jul 31 2003

See A057622 for the variant where "exactly" is replaced by "at least". See A055625 for the variant "congruent to 1 (mod 6)". - M. F. Hasler, Sep 03 2016

Links

• Giovanni Resta, Table of n, a(n) for n = 1..35 (terms < 4*10^14)
• J. K. Andersen, Consecutive Congruent Primes.

Crossrefs

Cf. A055623, A055624, A055625, A085516.

Programs

Mathematica

pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^6}];
sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
a[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 5} ..}]& ][[1, 1]];
Table[an = a[n]; Print[n, " ", an]; an, {n, 1, 13}] (* Jean-François Alcover, Nov 21 2018 *)

PARI

okchain(n, p) = {if ((precprime(p-1) % 6) == 5, return (0)); for (i=1, n, if ((p % 6) != 5, return (0)); p = nextprime(p+1);); if ((p % 6) == 5, 0, 1);}
a(n) = {p = 5; while (! okchain(n, p), p = nextprime(p+1)); p;} \\ Michel Marcus, Dec 17 2013

Extensions

a(9)-a(13), including correction of a(9)-a(10) from Reiner Martin, Jul 18 2001

a(14)-a(20) from Hugo Pfoertner, Jul 31 2003

a(21)-a(25) from Jens Kruse Andersen, May 30 2006

a(26) and beyond from Giovanni Resta, Aug 04 2013