A218046 Primes p such that 8p + 2r is a primorial for some r in A006512.

2, 11, 23, 83, 113, 131, 173, 191, 233, 239, 251, 263, 281, 293, 359, 419, 431, 449, 503, 641, 653, 659, 701, 719, 743, 761, 809, 821, 881, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1301, 1433, 1439, 1451, 1493, 1511, 1559, 1583, 1601, 1619

Offset: 1

Author

Michael G. Kaarhus, Oct 19 2012

Keywords

• nonn

Comments

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).

Each p is therefore associated with at least one primorial, and with a pair of twin primes.

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.

Examples

			8*2   + 2*7 = 5#
8*11  + 2*61 = 7#
8*23  + 2*13 = 7#
8*83  + 2*823 = 11#
8*113 + 2*14563 = 13#
8*131 + 2*254731 = 17#
8*173 + 2*463 = 11#
8*191 + 2*14251 = 13#
8*233 + 2*14083 = 13#
8*239 + 2*199 = 11#
8*251 + 2*151 = 11#
8*263 + 2*103 = 11#
8*281 + 2*31 = 11#
8*293 + 2*307444891294244533 = 47#
8*359 + 2*253819 = 17#
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..354
• Michael Kaarhus, Twin Prime Conjectures 1, 2 and 3, 2012, (PDF)

Programs

PARI

list(lim)={
    my(v=List(),P=3,q);
    forprime(p=5,lim,
        P*=p;
        forprime(t=2,min(lim, (P-2)\4),
            q=P-4*t;
            if(q%6==1 && ispseudoprime(q) && ispseudoprime(q-2), listput(v,t))
        )
    );
    vecsort(Vec(v),,8)
}; \\ Charles R Greathouse IV, Oct 23 2012

Extensions

Terms corrected by Charles R Greathouse IV, Oct 23 2012