A085516 -id:A085516 - cp's OEIS frontend

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

Labos Elemer, Jun 05 2000

nonn

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

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

Cf. A055623, A055624, A055625, A085516.

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 *)

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

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

A085515 Order of first occurrence of a sequence of exactly n consecutive primes of the form 6*k+1.

Original entry on oeis.org

1, 2, 3, 6, 4, 5, 7, 8, 10, 11, 9, 13, 12, 15, 16, 14, 17, 18, 21, 19, 20, 23, 22, 25, 24, 26, 31, 27, 28, 30, 29, 32, 34, 33, 35

Offset: 1

Hugo Pfoertner, Jul 31 2003

This sequence gives the index of A055625(n) after sorting by increasing magnitude.

			a(3)=3, a(4)=6, a(5)=4 because the first occurrence of exactly 6 consecutive primes of the form 6*k+1 (1741,1747,1753,1759,1777,1783) occurs after the first occurrence of 3 consecutive primes of this form (151,157,163) and before the first occurrence of 4 consecutive "6k+1"-primes (3049,3061,3067,3079).

J. K. Andersen, Consecutive Congruent Primes.
Hugo Pfoertner, Record producing runs of primes of the forms 6k+-1.
Hugo Pfoertner, FORTRAN function ISPRIM to check primality.

Cf. A055625, A085516 (sorted occurrence of first runs of "6k-1" primes).

Fortran
```
! Program given at link.
```

(* Program not suitable to compute a large number of terms. *)
pp = Table[{p = Prime[n], Mod[p, 6]}, {n, 10^7}];
sp = Split[pp, Mod[#1[[2]], 6] == Mod[#2[[2]], 6]&];
b[n_] := SelectFirst[sp, Length[#] == n && MatchQ[#, {{_Integer, 1} ..}]& ][[1, 1]];
Sort[Table[{b[n], n}, {n, 1, 16}]][[All, 2]] (* Jean-François Alcover, Nov 21 2018 *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 27 2006

a(32)-a(35) from Giovanni Resta, Aug 04 2013

cp's OEIS Frontend

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

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