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Least member p1 of prime octuplets (p1, p2, p3, ..., p8 = p1 + 26), the eight p's being consecutive primes.
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%I #40 Oct 26 2023 11:18:56

%S 11,17,1277,88793,113147,284723,855713,1146773,2580647,6560993,

%T 15760091,20737877,25658441,58208387,69156533,73373537,74266253,

%U 76170527,93625991,100658627,134764997,137943347,165531257,171958667

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

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

%C 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 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 Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets.htm">Prime k-tuplets</a>

%H Norman Luhn, <a href="http://www.pzktupel.de/smarchive.html">The smallest prime k-tuplets</a>, database of compressed files.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-TupleConjecture.html">k-Tuple Conjecture</a>

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

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

%Y Union of A022011, A022012 and A022013.

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

%K nonn

%O 1,1

%A _Frank Ellermann_, Dec 05 2001