A273995 - OEIS

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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

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OFFSET

1,1

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.)

LINKS

Table of n, a(n) for n=1..52.

EXAMPLE

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

PROG

(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)); }