1,1

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 by 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 counter-example n such that A002375(n)=0, i.e. 2n cannot be written as the sum of 2 primes.

Donovan Johnson, Table of n, a(n) for n=1..999

A137820(k+1) = min{ n>2 | A002375(n)/n < A002375(A137820(k))/A137820(k) }

(PARI) m=1; for(n=3, 10^4, n*m<=A002375(n)&next; m=A002375(n)/n; print1(n", "))

Sequence in context: A102733 A032712 A143100 this_sequence A049892 A063477 A129827

Adjacent sequences: A137817 A137818 A137819 this_sequence A137821 A137822 A137823

nonn

M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 23 2008