A332704 Record indices of the ratio A280008(n) / A002375(n) (Goldbach conjecture related).

3, 5, 33, 63, 165, 315, 255255

Offset: 1

Author

Craig J. Beisel, Jun 08 2020

Keywords

• nonn
• more

Comments

The sequence lists indices n for which A280008(n) / A002375(n) is less than all previous indices n > 2.

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.

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.

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.

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.

Crossrefs

Cf. A002375, A070826, A280008.

Programs

PARI

lastx=1; record=999; for(n=4, 1000, x=0; forprime(i=3, n, if(isprime(2*n-i), x=x+1; ); ); y=(x-lastx)/lastx; if(y
				

Formula

A332704(k+1) = min{ n>2 | A280008(n)/A002375(n) < A002375(A332704(k))/A280008(A332704(k)) }.