A284919 - OEIS

%I #99 Dec 23 2024 14:53:45

%S 0,2,4,6,28,38,52,58,62,68,74,80,82,88,94,98,112,118,122,124,128,136,

%T 146,148,152,158,164,166,172,178,182,184,188,190,206,208,212,214,218,

%U 220,224,238,242,244,248,250,256,262,268,278,284,290,292,296,298

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

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

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

%C 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

%C Conjecture: except for a(4)=6, all terms are coprime to 3. - _Bob Selcoe_, Apr 06 2017

%C 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

%C 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

%C 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

%C 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

%C 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

%H Robert G. Wilson v, <a href="/A284919/b284919.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from M. F. Hasler)

%H Claudio Meller and others, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2017-April/017433.html">New sequence</a>, SeqFan list, April 5, 2017. (Click "next" for subsequent contributions.)

%H Robert G. Wilson v, <a href="/A284919/a284919.pdf">Graph of all terms < 100000</a>

%F a(n) = 2*A284928(n). - _M. F. Hasler_, Apr 06 2017

%e 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

%t 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 *)

%o (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

%Y 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).

%K nonn

%O 1,2

%A _Claudio Meller_, Apr 05 2017