%I #31 Jun 28 2020 10:34:22

%S 22,46,58,146,344,362,526,1114,1781,2476,3097,3551,5131,5728,8504,

%T 10342,10907,10994,13321,13924,13984,18526,24776,26197,30728,40072,

%U 44656,44860,68707,70757,71684,76861,78461,89812,125903,181267,191771,227566,256849,278566,371428,379969

%N Integers k such that for all m>k, d(m)/m < d(k)/k where d(j) = Min_{p & q odd primes, 2*j = p+q, p <= q} (q-p)/2.

%C This sequence is related to a stronger form of Goldbach conjecture, and the formulation of the conjecture is the following.

%C Conjecture: Let

%C Gs(m) be the number of unordered pairs of odd prime numbers p and q such that an even number 2m can be written as the sum of p and q, or, Gs(m) = # { (p,q) | 2m = p+q }, where p <= q;

%C d be half of the minimum difference between q and p, or, d = min((q-p)/2);

%C r be the ratio of d/m, or, r = d/m; and

%C a(n) be the n-th number such that r = d/m is decreasing, or, r(m) < r(a(n+1)), if a(n) <= m < a(n+1), where n = 0, 1, 2, 3, .... and a(0)=3.

%C The conjecture states that

%C Gs(m) >= 1 if r(m) >= r(a(n+1)) where a(n) <= m < a(n+1).

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

%e For even numbers 2m >= 6 (6 is the smallest even number that can be written as the sum of two odd primes), the list of m is:

%e m = {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, ...}.

%e The corresponding values of d and r, according to the definition, are given in the following two lists:

%e d = {0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, ...}, and

%e r = {0, 1/4, 0, 1/6, 0, 3/8, 2/9, 3/10, 0, 1/12, 0, 3/14, 2/15, 3/16, 0, 1/18, 0, 3/20, 2/21, 9/22, 0, 5/24, 6/25, 3/26, 4/27, 9/28, 0, 1/30, 0, 9/32, 4/33, 3/34, 6/35, 5/36, 0, 9/38, 2/39, 3/40, 0, 1/42, 0, 3/44, 2/45, 15/46, ...}.

%e In the list of r, the first number that is bigger than all the preceding numbers is r = 9/22, which is corresponding to the number m = 22 in the list of m. Therefore, the first number of the sequence is 22, or a(1) = 22.

%e In the range of (9/22, 15/46], r= 15/46 is the biggest number. Since r = 15/46 corresponds to m = 46, the 2nd number of the sequence is 46, or a(2) = 46.

%e The first number in the list of m, 3, is defined as the zeroth term of the sequence, or a(0) = 3.

%o (PARI) mindiff(n) = {forstep(k=n/2, 1, -1, if (isprime(k) && isprime(n-k), return(n-2*k)););}

%o upto(n) = res=List(); r=0;forstep(i=n, 1, -1, c= mindiff(2*i) / (2*i); if(c>r,r=c;listput(res,i))); Vecrev(Vec(res)) \\ _David A. Corneth_, Jun 02 2020

%Y Cf. A002375, A066285.

%K nonn

%O 1,1

%A _Ya-Ping Lu_, May 30 2020