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A137820 Record indices of the ratio A002375(n) / n (Goldbach conjecture related). 2
3, 4, 6, 14, 16, 19, 31, 34, 64, 163, 166, 199, 316, 496, 706, 859, 1024, 1126, 1321, 1336, 2206, 2539, 2644, 2719, 2734, 2974, 3646, 3754, 3931, 4021, 4801, 6826, 7894, 8431, 8506, 9109, 9623, 9904, 10084, 10174, 10321, 10639, 11749, 11839, 13894, 13954, 16174 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

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.

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.

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.

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

LINKS

Donovan Johnson, Table of n, a(n) for n = 1..999

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

FORMULA

A137820(k+1) = Min_{ n>2 | A002375(n)/n < A002375(A137820(k))/A137820(k) }.

PROG

(PARI) m=1; for(n=3, 10^4, n*m<=A002375(n)&next; m=A002375(n)/n; print1(n", "))

CROSSREFS

Sequence in context: A327584 A180859 A271618 * A049892 A063477 A168219

Adjacent sequences:  A137817 A137818 A137819 * A137821 A137822 A137823

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 23 2008

STATUS

approved

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Last modified May 17 05:02 EDT 2021. Contains 343964 sequences. (Running on oeis4.)