A054203 a(n) is the smallest start of a run of exactly n+1 consecutive primes with n (not necessarily equal) prime differences, each divisible by 6.

23, 47, 251, 1889, 1741, 19471, 118801, 498259, 148531, 406951, 1820111, 2339041, 40727657, 19725473, 73451737, 232301497, 400414121, 1057859471, 489144599, 1444257673, 766319189, 24061965043, 87996684091, 21549657539, 141116164769, 140432294381, 437339303279

Offset: 1

Author

Labos Elemer, May 17 2000

Keywords

• nonn

Comments

This is a "modular arithmetic progression" of successive primes, modulo 6.

Examples

			For n = 1: a(1) = 23 is followed by a difference 6 to give 29, a prime.
For n = 5 a(5) = 1741 is followed by differences {6, 6, 6, 18, 6} and results in {1741, 1747, 1753, 1759, 1777, 1783} consecutive prime sequence.
For n = 10: a(10) = 406951 is prime prime is followed by {18, 12, 12, 30, 24, 12, 24, 36, 18, 12} consecutive differences pattern.
		

Links

• Index entries for sequences related to primes in arithmetic progressions.

Crossrefs

Cf. A001223, A033451, A052243, A052239.

Programs

PARI

list(len) = {my(s = vector(len), v = [], prv = 2, c = 0, i, q, d); forprime(p = 3, , d = p - prv; if(d % 6, if(q > 0, i = #v; if(i > 0 && i <= len && s[i] == 0, s[i] = q; c++)); v = [], if(#v == 0, q = prv); v = concat(v, p)); prv = p; if(c == len, break)); s;} \\ Amiram Eldar, Mar 11 2025

Extensions

a(11)-a(21) from Sean A. Irvine, Jan 25 2022

a(8) corrected, a(22)-a(27) added, and name clarified by Amiram Eldar, Mar 11 2025