A273995 Even numbers with a unique representation as the difference of two primes, each of which is a member of a pair of twin primes, and one of which is smaller than the even number under consideration.

4, 6, 20, 34, 46, 50, 74, 82, 86, 202, 206, 214, 218, 244, 248, 256, 260, 352, 356, 382, 386, 454, 472, 476, 524, 562, 604, 608, 664, 668, 724, 728, 772, 776, 982, 986, 1162, 1166, 1192, 1196, 1552, 1556, 1672, 1676, 2872, 2876, 3082, 3086, 6232, 6236, 6892, 6896

Offset: 1

Author

Thomas Curtright, Jun 06 2016

Keywords

• nonn

Comments

For the sequence to be infinite there must be an infinite number of twin prime pairs.

Can any even number n > 2 be so written (perhaps not uniquely) as the difference of two (unrelated) twins, one of which is smaller than n? (T. S. Van Kortryk conjectures there are, if any, only a finite number of even integers such that this is not true.)

Examples

			For even n with 4 <= n <= 100, all have at least one representation as the difference of two primes, each of which is a member of a pair of twin primes, but the following have only one such representation, and so belong to the sequence:
     4 =   7 -  3
     6 =  11 -  5
    20 =  31 - 11
    34 =  41 -  7
    46 =  59 - 13
    50 =  61 - 11
    74 = 103 - 29
    82 = 101 - 19
    86 = 103 - 17
		

Crossrefs

Cf. A007534.

Programs

PARI

istwin(p) = isprime(p+2) || isprime(p-2);
isok(n) = {my(nb = 0); forprime(p=3, n, if (isprime(n+p) && istwin(p) && istwin(n+p), nb++);); if (nb == 1, return (1));}
lista(nn) = forstep(n=4, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Jun 07 2016

Extensions

More terms from Michel Marcus, Jun 07 2016