cp's OEIS Frontend

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

%I A137820 #34 Jun 24 2025 20:53:28
%S A137820 3,4,6,14,16,19,31,34,64,163,166,199,316,496,706,859,1024,1126,1321,
%T A137820 1336,2206,2539,2644,2719,2734,2974,3646,3754,3931,4021,4801,6826,
%U A137820 7894,8431,8506,9109,9623,9904,10084,10174,10321,10639,11749,11839,13894,13954,16174
%N A137820 Record indices of the ratio A002375(n) / n (Goldbach conjecture related).
%C A137820 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.
%C A137820 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.
%C A137820 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.
%C A137820 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.
%C A137820 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.
%C A137820 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
%H A137820 Donovan Johnson, <a href="/A137820/b137820.txt">Table of n, a(n) for n = 1..999</a>
%H A137820 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A137820 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A137820 A137820(k+1) = min { n>2 | A002375(n)/n < A002375(A137820(k))/A137820(k) }.
%o A137820 (PARI) m=1;for(n=3,10^4,n*m<=A002375(n)&&next;m=A002375(n)/n;print1(n", "))
%Y A137820 Cf. A002375 (number of ways to write 2n as sum of two primes).
%K A137820 nonn
%O A137820 1,1
%A A137820 _M. F. Hasler_, Feb 23 2008