OFFSET
1,1
COMMENTS
The sequence lists indices n for which A002375(n)/n is less than for all previous indices n > 2, or equivalently, assuming that A002375(n) > 0 for all n > 2 (Goldbach conjecture), values for which n/A002375(n) is greater than for all previous indices n > 2.
We do not consider indices n=1 and n=2, for which the sequence A002375(n) (= number of prime {p,q} such that 2n=p+q) is zero.
Note also that A045917=A002375 except for n=2; since we exclude n < 3, one can equivalently replace one of these two with the other in the definition.
In A002375, an upper bound for A002375(n) is given; however, the Goldbach conjecture is A002375(n) > 0 for all n > 2, thus rather connected to the question of a lower bound. This sequence lists values of n for which A002375(n) is particularly low.
If the conjecture is wrong, then this sequence A137820 is finite: It will end with the counterexample n such that A002375(n)=0, i.e., 2n cannot be written as the sum of 2 primes.
Conjecture: All terms of this sequence are of the form 2^i, 2^i*p, or 2^i*p*q where i>=0 and p and q not necessarily distinct odd primes. - Craig J. Beisel, Jun 15 2020
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..999
Wikipedia, Goldbach's conjecture
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Feb 23 2008
STATUS
approved