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

%I A065706 #41 Feb 16 2025 08:32:45
%S A065706 11,17,1277,88793,113147,284723,855713,1146773,2580647,6560993,
%T A065706 15760091,20737877,25658441,58208387,69156533,73373537,74266253,
%U A065706 76170527,93625991,100658627,134764997,137943347,165531257,171958667
%N A065706 Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.
%C A065706 3 patterns for 8-tuplets: 11010011001011, 11011010011001 and v.v.
%C A065706 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
%H A065706 Harry J. Smith and Dana Jacobsen, <a href="/A065706/b065706.txt">Table of n, a(n) for n = 1..18123</a> [first 100 terms from Harry J. Smith]
%H A065706 Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets.htm">Prime k-tuplets</a>
%H A065706 Norman Luhn, <a href="http://www.pzktupel.de/smarchive.html">The smallest prime k-tuplets</a>, database of compressed files.
%H A065706 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>
%e A065706 a(3) = 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303 = 1277+26 are primes.
%o A065706 (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
%o A065706 (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
%Y A065706 11 = A065688(8), 26 = A008407(8), 8 = A023193(26+1), octets in A066082 are another (not minimal) constellation of 8 primes.
%Y A065706 Union of A022011, A022012 and A022013.
%Y A065706 See A257124 (prime septuplets) with an overview of prime k-tuplets.
%K A065706 nonn
%O A065706 1,1
%A A065706 _Frank Ellermann_, Dec 05 2001