%I #53 Apr 25 2023 01:04:27
%S 3,5,33,63,165,315,255255
%N Record indices of the ratio A280008(n) / A002375(n) (Goldbach conjecture related).
%C The sequence lists indices n for which A280008(n) / A002375(n) is less than all previous indices n > 2.
%C We do not consider indices n=1 and n=2, for which the sequence A002375(n) (= number of odd primes {p,q} such that 2n=p+q) is zero.
%C If the Goldbach conjecture is false, then this sequence is finite. It will end with n such that A280008(n) / A002375(n) = 1 since no further terms could achieve less than this value.
%C If the Goldbach conjecture is true, then this sequence may be finite or infinite. The ratio A280008(n) / A002375(n) has a lower bound greater than 1 and the value of this ratio for record indices approaches the lower bound.
%C It is known that this sequence has additional terms beyond a(7) = 17#/2 = 255255 = A070826(7) since A280008(255255) / A002375(255255) = 0.76119 and for A070826(10) = 20#/2 = 3234846615 we have A280008(3234846615) / A002375(3234846615) = 0.78989.
%F A332704(k+1) = min{ n>2  A280008(n)/A002375(n) < A002375(A332704(k))/A280008(A332704(k)) }.
%o (PARI) lastx=1; record=999; for(n=4, 1000, x=0; forprime(i=3, n, if(isprime(2*ni), x=x+1; ); ); y=(xlastx)/lastx; if(y<record, record=y; print1(n1, ", " ); ); lastx=x; );
%Y Cf. A002375, A070826, A280008.
%K nonn,more
%O 1,1
%A _Craig J. Beisel_, Jun 08 2020
