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A273995
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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.
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0
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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
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1,1
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COMMENTS
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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.)
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LINKS
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EXAMPLE
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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
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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