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A307542 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^2.

%I A307542 #20 Apr 27 2019 17:46:51
%S A307542 4,6,8,10,12,18,24,28,30,38,54,98,122,124,126,128,220,302,308,332,346,
%T A307542 368,488,556,854,908,962,968,992,1144,1150,1274,1354,1360,1362,1382,
%U A307542 1408,1424,1532,1768,1856,1928,2078,2188,2200,2438,2512,2530,2618,2642,3458,3526,3818,3848
%N A307542 Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and (k - p) < p^2.
%C A307542 This sequence is an extension of A244408. It is equivalent to "Even numbers 2n such that the smallest prime p satisfying p+q=2n (p, q prime, p<=q) also satisfies p^2+p>2n." If p satisfies additionally p^2 < 2n the corresponding even numbers do not belong to A244408. These numbers are 10, 28, 54, 124, 368, 968, 3526. It is conjectured that a(81)=63274 is the last term. There are no more terms below 4*10^18.
%H A307542 Corinna Regina Böger, <a href="/A307542/b307542.txt">Table of n, a(n) for n = 1..81</a>
%e A307542 10 = 3 + 7, 3^2 = 9 < 10 and 9 > 7 = q, therefore it is in this sequence.
%p A307542 isS := proc(n) local p; for p from 2 while p^2 < (n-p) do
%p A307542 if isprime(p) and isprime(n-p) then return false fi od; true end:
%p A307542 isa := n -> irem(n, 2) = 0 and isS(n): select(isa, [$4..3848]); # _Peter Luschny_, Apr 26 2019
%t A307542 Select[Range[4, 4000, 2], #2 > Sqrt@ #1 & @@ SelectFirst[IntegerPartitions[#, {2}], AllTrue[#, PrimeQ] &] &] (* _Michael De Vlieger_, Apr 21 2019 *)
%o A307542 (PARI) noSpecialGoldbach(n) = forprime(p=2, n/2, if(p^2+p<n && isprime(n-p), return(0))); 1
%o A307542 is(n) = n>2 && n%2 == 0 && noSpecialGoldbach(n)
%Y A307542 Cf. A244408, A093161, A279040.
%K A307542 nonn
%O A307542 1,1
%A A307542 _Corinna Regina Böger_, Apr 14 2019