A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime.

0, 2, 4, 6, 28, 38, 52, 58, 62, 68, 74, 80, 82, 88, 94, 98, 112, 118, 122, 124, 128, 136, 146, 148, 152, 158, 164, 166, 172, 178, 182, 184, 188, 190, 206, 208, 212, 214, 218, 220, 224, 238, 242, 244, 248, 250, 256, 262, 268, 278, 284, 290, 292, 296, 298

Offset: 1

Author

Claudio Meller, Apr 05 2017

Keywords

• nonn

Comments

Or, even integers k such that k + p is composite for all primes p, q with p + q = k.

The two initial terms vacuously satisfy the definition, but all even numbers >= 4 are the sum of two primes, according to the Goldbach conjecture.

All odd numbers except for numbers m such that m-2 and m+2 are prime (= A087679) would satisfy the definition. - M. F. Hasler, Apr 05 2017

Conjecture: except for a(4)=6, all terms are coprime to 3. - Bob Selcoe, Apr 06 2017

If E is an even number not divisible by 3, then E is in the sequence unless E-3 and at least one of E+3 and 2E-3 are prime. - Robert Israel, Apr 10 2017

Consider a subsequence with the additional condition: n+odd part of p-1 is composite (for example, for p=19 it is 9). I found that this subsequence begins 0,2,118 and up to 300000 Peter J. C. Moses found only more one term 868. Is this subsequence finite? - Vladimir Shevelev, Apr 12 2017

One can compare the theoretical maxima with the actual sequence numbers of terms. Doing this at powers of 10, we see at powers {2,3,4,5,6} the ratio progression {2.33, 1.51, 1.25, 1.15, 1.096}. This implies that the excluded even numbers become increasingly rare (those coprime to 3). - Bill McEachen, Apr 17 2017

From Robert Israel's comment and the distribution of primes, the proportion of even numbers not divisible by 3 that are in the sequence tends to 1. - Peter Munn, Apr 23 2017

Moreover, If n is not divisible by 3 and 2*n - 3 is composite, then 2*n+p is also composite. Indeed, for these 2*n all primes p such that 2*n-p is prime are in the interval (3, 2*n-3). Then either 2*n-p or 2*n+p should be divisible by 3, but 2*n-p is a prime > 3. So 2*n+p is composite and 2*n is in the sequence. - Vladimir Shevelev, Apr 28 2017

Examples

			k=28 is in the sequence because 5+23 = 28 and 11+17 = 28, and 28 + {5,11,17,23} are composite; k=26 is not in the sequence because 3+23 = 26, 7+19 = 26 and 13+13 = 26, but 26+3 = 29 (prime).  - _Bob Selcoe_, Apr 06 2017
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 1000 terms from M. F. Hasler)
• Claudio Meller and others, New sequence, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)
• Robert G. Wilson v, Graph of all terms < 100000

Crossrefs

Cf. A284928 (terms/2), A002375 (number of decompositions p + q = 2k), A020481 (least p: p + q = 2k), A277688 (an analog for decompositions odd k as 2p+q).

Programs

Mathematica

fQ[n_] :=  Select[Select[Prime@Range@PrimePi@n, PrimeQ[n - #] &],    PrimeQ[n + #] &] == {}; Select[2 Range[0, 150], fQ] (* Robert G. Wilson v, Apr 05 2017 *)

PARI

is(n)=!bittest(n,0)&&!forprime(p=2,n\2, isprime(n-p) && (isprime(n+p) || isprime(2*n-p)) && return) \\ Charles R Greathouse IV and M. F. Hasler, Apr 05 2017

Formula

a(n) = 2*A284928(n). - M. F. Hasler, Apr 06 2017