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.

4, 6, 8, 10, 12, 18, 24, 28, 30, 38, 54, 98, 122, 124, 126, 128, 220, 302, 308, 332, 346, 368, 488, 556, 854, 908, 962, 968, 992, 1144, 1150, 1274, 1354, 1360, 1362, 1382, 1408, 1424, 1532, 1768, 1856, 1928, 2078, 2188, 2200, 2438, 2512, 2530, 2618, 2642, 3458, 3526, 3818, 3848

Offset: 1

Author

Corinna Regina Böger, Apr 14 2019

Keywords

• nonn

Comments

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.

Examples

			10 = 3 + 7, 3^2 = 9 < 10 and 9 > 7 = q, therefore it is in this sequence.
		

Links

• Corinna Regina Böger, Table of n, a(n) for n = 1..81

Crossrefs

Cf. A244408, A093161, A279040.

Programs

Maple

isS := proc(n) local p; for p from 2 while p^2 < (n-p) do
if isprime(p) and isprime(n-p) then return false fi od; true end:
isa := n -> irem(n, 2) = 0 and isS(n): select(isa, [$4..3848]); # Peter Luschny, Apr 26 2019

Mathematica

Select[Range[4, 4000, 2], #2 > Sqrt@ #1 & @@ SelectFirst[IntegerPartitions[#, {2}], AllTrue[#, PrimeQ] &] &] (* Michael De Vlieger, Apr 21 2019 *)

PARI

noSpecialGoldbach(n) = forprime(p=2, n/2, if(p^2+p2 && n%2 == 0 && noSpecialGoldbach(n)