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A332704 Record indices of the ratio A280008(n) / A002375(n) (Goldbach conjecture related). 1
3, 5, 33, 63, 165, 315, 255255 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
FORMULA
A332704(k+1) = min{ n>2 | A280008(n)/A002375(n) < A002375(A332704(k))/A280008(A332704(k)) }.
PROG
(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<record, record=y; print1(n-1, ", " ); ); lastx=x; );
CROSSREFS
Sequence in context: A372718 A103010 A225866 * A263295 A222484 A354849
KEYWORD
nonn,more
AUTHOR
Craig J. Beisel, Jun 08 2020
STATUS
approved

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Last modified July 14 02:38 EDT 2024. Contains 374291 sequences. (Running on oeis4.)