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%I #37 May 13 2013 01:54:22
%S 2,11,23,83,113,131,173,191,233,239,251,263,281,293,359,419,431,449,
%T 503,641,653,659,701,719,743,761,809,821,881,911,953,1013,1019,1031,
%U 1049,1103,1223,1229,1289,1301,1433,1439,1451,1493,1511,1559,1583,1601,1619
%N Primes p such that 8p + 2r is a primorial for some r in A006512.
%C The primes p in this sequence satisfy b#/2 = 4p + r, where p is a prime, b# is a primorial, and r is the second of the twin prime pair (r-2, r).
%C Each p is therefore associated with at least one primorial, and with a pair of twin primes.
%C The empirical evidence suggests that each twin prime pair is associated with at least one p, and each p with a twin prime pair. I conjecture that this sequence (and therefore the sequence of twin primes) is infinite.
%H Charles R Greathouse IV, <a href="/A218046/b218046.txt">Table of n, a(n) for n = 1..354</a>
%H Michael Kaarhus, <a href="http://www.christaboveme.com/pri/conj-ii-d.pdf">Twin Prime Conjectures 1, 2 and 3</a>, 2012, (PDF)
%e 8*2 + 2*7 = 5#
%e 8*11 + 2*61 = 7#
%e 8*23 + 2*13 = 7#
%e 8*83 + 2*823 = 11#
%e 8*113 + 2*14563 = 13#
%e 8*131 + 2*254731 = 17#
%e 8*173 + 2*463 = 11#
%e 8*191 + 2*14251 = 13#
%e 8*233 + 2*14083 = 13#
%e 8*239 + 2*199 = 11#
%e 8*251 + 2*151 = 11#
%e 8*263 + 2*103 = 11#
%e 8*281 + 2*31 = 11#
%e 8*293 + 2*307444891294244533 = 47#
%e 8*359 + 2*253819 = 17#
%o (PARI) list(lim)={
%o my(v=List(),P=3,q);
%o forprime(p=5,lim,
%o P*=p;
%o forprime(t=2,min(lim, (P-2)\4),
%o q=P-4*t;
%o if(q%6==1 && ispseudoprime(q) && ispseudoprime(q-2), listput(v,t))
%o )
%o );
%o vecsort(Vec(v),,8)
%o }; \\ _Charles R Greathouse IV_, Oct 23 2012
%K nonn
%O 1,1
%A _Michael G. Kaarhus_, Oct 19 2012
%E Terms corrected by _Charles R Greathouse IV_, Oct 23 2012
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