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%I #30 Aug 06 2020 22:26:01
%S 3,4,6,14,16,19,31,34,64,163,166,199,316,496,706,859,1024,1126,1321,
%T 1336,2206,2539,2644,2719,2734,2974,3646,3754,3931,4021,4801,6826,
%U 7894,8431,8506,9109,9623,9904,10084,10174,10321,10639,11749,11839,13894,13954,16174
%N Record indices of the ratio A002375(n) / n (Goldbach conjecture related).
%C 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 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 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 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 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 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 Donovan Johnson, <a href="/A137820/b137820.txt">Table of n, a(n) for n = 1..999</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F A137820(k+1) = Min_{ n>2 | A002375(n)/n < A002375(A137820(k))/A137820(k) }.
%o (PARI) m=1;for(n=3,10^4,n*m<=A002375(n)&next;m=A002375(n)/n;print1(n", "))
%K nonn
%O 1,1
%A _M. F. Hasler_, Feb 23 2008
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