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A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).

%I A279040 #24 May 01 2019 09:12:56
%S A279040 4,6,8,10,12,14,16,18,24,28,30,36,38,42,48,54,60,68,80,90,96,98,122,
%T A279040 124,126,128,148,150,190,192,208,210,212,220,222,224,302,306,308,326,
%U A279040 330,332,346,368,398,418,458,488,518,538,540,542,556,640,692,710,796,854,908,962,968,992,1006
%N A279040 Even numbers 2k such that the smallest prime p satisfying p+q=2k (q prime) is greater than or equal to sqrt(k).
%C A279040 a(n) is an extension of A244408.
%C A279040 It is conjectured that a(230) = 503222 is the last term. Oliveira e Silva's work shows that there are no more terms below 4*10^18.
%C A279040 The sequence definition is equivalent to: "Even integers k such that there exists a prime p with p = min{q: q prime and (k - q) prime} and k < 2*p^2" and therefore this is a member of the EGN- family (Cf. A307782). - _Corinna Regina Böger_, May 01 2019
%H A279040 Corinna Regina Böger, <a href="/A279040/b279040.txt">Table of n, a(n) for n = 1..230</a>
%H A279040 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>
%H A279040 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%e A279040 The smallest prime for 42 is 5 with 5 > sqrt(21), but not smaller than sqrt(42), and therefore 42 does not belong to A244408. The smallest prime for 38 is 7, and 7 >= sqrt(38), and therefore 38 also belongs to A244408.
%t A279040 Select[Range[4, 1006, 2], Function[n, Select[#, PrimeQ@ Last@ # &][[1, 1]] >= Sqrt[n/2] &@ Map[{#, n - #} &, Prime@ Range@ PrimePi@ n]]] (* _Michael De Vlieger_, Dec 06 2016 *)
%o A279040 (PARI) isok(n) = forprime(p=2, n, if (isprime(n-p), if (p >= sqrt(n/2), return(1), return(0))));
%o A279040 lista(nn) = forstep(n=2, nn, 2, if (isok(n), print1(n, ", "))) \\ _Michel Marcus_, Dec 04 2016
%Y A279040 Cf. A002373, A020481, A025018, A025019, A093161, A244408, A307542, A307782
%K A279040 nonn
%O A279040 1,1
%A A279040 _Corinna Regina Böger_, Dec 04 2016