A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.

11, 17, 1277, 88793, 113147, 284723, 855713, 1146773, 2580647, 6560993, 15760091, 20737877, 25658441, 58208387, 69156533, 73373537, 74266253, 76170527, 93625991, 100658627, 134764997, 137943347, 165531257, 171958667

Offset: 1

Author

Frank Ellermann, Dec 05 2001

Keywords

• nonn

Comments

3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.

See A022011, A022012 and A022013 for the three different possible patterns. The sequence is conjectured to be infinite, although it is not even proved that there are infinitely many twin primes (p1, p2 = p1+2). - M. F. Hasler, May 02 2015

Examples

			a(3) = 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303 = 1277+26 are primes.
		

Links

• Harry J. Smith and Dana Jacobsen, Table of n, a(n) for n = 1..18123 [first 100 terms from Harry J. Smith]
• Tony Forbes and Norman Luhn, Prime k-tuplets
• Norman Luhn, The smallest prime k-tuplets, database of compressed files.
• Eric Weisstein's World of Mathematics, k-Tuple Conjecture

Crossrefs

11 = A065688(8), 26 = A008407(8), 8 = A023193(26+1), octets in A066082 are another (not minimal) constellation of 8 primes.

Union of A022011, A022012 and A022013.

See A257124 (prime septuplets) with an overview of prime k-tuplets.

Programs

PARI

{ n=0; p1=2; p8=19; for (m=1, 10^12, p1=nextprime(p1+1); p8=nextprime(p8+1); if (p8 - p1 == 26, write("b065706.txt", n++, " ", p1); if (n==100, return)) ) } \\ Harry J. Smith, Oct 26 2009

Perl

use ntheory ":all"; my($s,$e,$i,%h)=(1,1e10,0); undef @h{sieve_prime_cluster($s,$e,2,6,8,12,18,20,26), sieve_prime_cluster($s,$e,2,6,12,14,20,24,26), sieve_prime_cluster($s,$e,6,8,14,18,20,24,26)}; say ++$i," $" for sort {$a<=>$b} keys %h; # _Dana Jacobsen, Oct 10 2015