

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
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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 m2 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 E3 and at least one of E+3 and 2E3 are prime.  Robert Israel, Apr 10 2017
Consider a subsequence with the additional condition: n+odd part of p1 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*np is prime are in the interval (3, 2*n3). Then either 2*np or 2*n+p should be divisible by 3, but 2*np 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)<n/2 do b:=nithprime(a); if isprime(b) then
if isprime(n+ithprime(a)) or isprime(n+b) then ok:=0; break; fi; fi; a:=a+1; od;
if ok=1 then print(n); fi; od; end: P(3*10^2); # Paolo P. Lava, Apr 06 2017


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 *)


PROG

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


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).
Sequence in context: A195333 A106274 A204661 * A279419 A077633 A006933
Adjacent sequences: A284916 A284917 A284918 * A284920 A284921 A284922


KEYWORD

nonn


AUTHOR

Claudio Meller, Apr 05 2017


STATUS

approved



