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A274057 Even numbers n that cannot be written as the difference of two primes, each of which is one of a pair of twin primes, and one of which is < n.

2, 112, 114, 116, 364, 366, 368, 394, 396, 398, 484, 486, 488, 694, 696, 698, 934, 936, 938, 1402, 1404, 1406

Offset: 1

Thomas Curtright, Jun 08 2016

nonn

There are no more terms less than 10^5.

There are no more terms less than 10^8. Probably there are no more terms; a Cramér model predicts that the chance of more terms existing is less than 1 in 10^2000. - Charles R Greathouse IV, Jun 17 2016

is(n)=my(p=2); forprime(q=3,n+1,if(q-p!=2,p=q;next); if(isprime(n+p) && (isprime(n+p-2) || isprime(n+p+2)),return(0)); if(isprime(n+q) && (isprime(n+q-2) || isprime(n+q+2)) && qCharles R Greathouse IV, Jun 17 2016

A274001 Even numbers with a unique resolution as the sum of two primes, each of which has a twin.

Original entry on oeis.org

6, 8, 12, 28, 40, 52, 56, 68, 124, 128, 136, 172, 176, 188, 226, 262, 266, 304, 308, 394, 396, 398, 412, 416, 442, 446, 484, 488, 544, 548, 556, 560, 608, 634, 638, 668, 682, 686, 694, 696, 698, 724, 728, 736, 740, 754, 758, 772, 776, 802, 806, 874, 878, 934

Offset: 1

Thomas Curtright, Jun 06 2016

nonn

The sequence is infinite only if the number of twin primes is infinite.

Note that not all even integers can be written as the sum of two twins (e.g. 94, 96, 98,...).

			6 = 3 + 3 is an element since (3,5) are twins, as is 8 = 5 + 3.
10 = 7 + 3 = 5 + 5 is not an element, since it is not uniquely resolved, even though the two resolutions both involve primes with twins.

Cf. A129363, A007534.

ok[n_] := 1 == Length@ IntegerPartitions[n, {2}, Select[Prime@ Range@ PrimePi@ n, Or @@ PrimeQ[# + {-2, 2}] &]]; Select[2 Range[500], ok] (* Giovanni Resta, Jun 06 2016 *)

a(7)-a(54) from Giovanni Resta, Jun 06 2016

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.

Original entry on oeis.org

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

Thomas Curtright, Jun 06 2016

nonn

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.)

			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

Cf. A007534.

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

More terms from Michel Marcus, Jun 07 2016

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User: Thomas Curtright

A274057 Even numbers n that cannot be written as the difference of two primes, each of which is one of a pair of twin primes, and one of which is < n.

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A274001 Even numbers with a unique resolution as the sum of two primes, each of which has a twin.

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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.

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PARI

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