A137820 Record indices of the ratio A002375(n) / n (Goldbach conjecture related).

3, 4, 6, 14, 16, 19, 31, 34, 64, 163, 166, 199, 316, 496, 706, 859, 1024, 1126, 1321, 1336, 2206, 2539, 2644, 2719, 2734, 2974, 3646, 3754, 3931, 4021, 4801, 6826, 7894, 8431, 8506, 9109, 9623, 9904, 10084, 10174, 10321, 10639, 11749, 11839, 13894, 13954, 16174

Offset: 1

Author

M. F. Hasler, Feb 23 2008

Keywords

• nonn

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
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A002375 (number of ways to write 2n as sum of two primes).

Programs

PARI

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

Formula

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