A279040 - OEIS

%I #24 May 01 2019 09:12:56

%S 4,6,8,10,12,14,16,18,24,28,30,36,38,42,48,54,60,68,80,90,96,98,122,

%T 124,126,128,148,150,190,192,208,210,212,220,222,224,302,306,308,326,

%U 330,332,346,368,398,418,458,488,518,538,540,542,556,640,692,710,796,854,908,962,968,992,1006

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

%C a(n) is an extension of A244408.

%C 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 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 Corinna Regina Böger, <a href="/A279040/b279040.txt">Table of n, a(n) for n = 1..230</a>

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%e 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 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 (PARI) isok(n) = forprime(p=2, n, if (isprime(n-p), if (p >= sqrt(n/2), return(1), return(0))));

%o lista(nn) = forstep(n=2, nn, 2, if (isok(n), print1(n, ", "))) \\ _Michel Marcus_, Dec 04 2016

%Y Cf. A002373, A020481, A025018, A025019, A093161, A244408, A307542, A307782

%K nonn

%O 1,1

%A _Corinna Regina Böger_, Dec 04 2016