A274189 - OEIS

%I #35 Mar 15 2021 01:53:54

%S 34,46,50,66,74,78,86,138,142,160,162,168,170,174,176,178,180,184,186,

%T 194,202,204,206,234,236,238,240,242,246,252,254,264,270,276,282,284,

%U 290,294,296,298,300,310,320,324,328,334,348,354,364,366,370,372,376,378,384,386,390,392,396,400

%N Even numbers 2n that satisfy an extended Goldbach conjecture: They have at least one Goldbach partition 2n = p + q (p <= q; p, q prime) that satisfies p <= sqrt(n), at least one with sqrt(n) < p <= sqrt(2n) and at least one with p > sqrt(2n).

%C This sequence contains all even numbers that are not in A279040 or in A273457. I have verified numerically for all even numbers 4 < 2n < 4*10^10 that a Goldbach partition with the additional condition p > sqrt(2n) exists. It is conjectured that a(n) = 2*(n+12987) for all n > 7838315. If this conjecture is true, all even numbers 2n > 15702604 have all three types of Goldbach partitions and therefore satisfy the "extended Goldbach conjecture".

%H Corinna Regina Böger, <a href="/A274189/b274189.txt">Table of n, a(n) for n = 1..100000</a>

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

%e a(1) = 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. Since 3 < sqrt(17) < 5 < sqrt(34) < 11 < 17, all three types of Goldbach partitions exist for 34.

%o (PARI) GoldbachRange(n,mn,mx)=forprime(p=mn,mx, if(isprime(n-p), return(1))); 0

%o is(n)=n%2==0 && GoldbachRange(n, 2, sqrtint(n/2)) && GoldbachRange(n, sqrtint(n/2-1)+1, sqrtint(n)) && GoldbachRange(n, sqrtint(n-1)+1, n/2) \\ _Charles R Greathouse IV_, Dec 16 2016

%Y Cf. A002373, A020481, A025018, A025019, A244408, A273457, A279040.

%K nonn

%O 1,1

%A _Corinna Regina Böger_, Dec 11 2016