A273995 - OEIS

%I #28 Jul 14 2021 10:04:12

%S 4,6,20,34,46,50,74,82,86,202,206,214,218,244,248,256,260,352,356,382,

%T 386,454,472,476,524,562,604,608,664,668,724,728,772,776,982,986,1162,

%U 1166,1192,1196,1552,1556,1672,1676,2872,2876,3082,3086,6232,6236,6892,6896

%N 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.

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

%C 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.)

%e 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:

%e      4 =   7 -  3

%e      6 =  11 -  5

%e     20 =  31 - 11

%e     34 =  41 -  7

%e     46 =  59 - 13

%e     50 =  61 - 11

%e     74 = 103 - 29

%e     82 = 101 - 19

%e     86 = 103 - 17

%o (PARI) istwin(p) = isprime(p+2) || isprime(p-2);

%o isok(n) = {my(nb = 0); forprime(p=3, n, if (isprime(n+p) && istwin(p) && istwin(n+p), nb++);); if (nb == 1, return (1));}

%o lista(nn) = forstep(n=4, nn, 2, if (isok(n), print1(n, ", "))); \\ _Michel Marcus_, Jun 07 2016

%Y Cf. A007534.

%K nonn

%O 1,1

%A _Thomas Curtright_, Jun 06 2016

%E More terms from _Michel Marcus_, Jun 07 2016