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 A284919 Even integers E such that there is no prime p < E with E - p and E + p both prime. 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 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 FORMULA a(n) = 2*A284928(n). - M. F. Hasler, Apr 06 2017 EXAMPLE 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 MAPLE with(numtheory):P:=proc(q) local a, b, k, ok, n; for n from 0 by 2 to q do a:=2; ok:=1; while ithprime(a)

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Last modified August 9 13:39 EDT 2020. Contains 336323 sequences. (Running on oeis4.)