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A137820 Record indices of the ratio A002375(n) / n (Goldbach conjecture related). 1
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 by 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 counter-example n such that A002375(n)=0, i.e. 2n cannot be written as the sum of 2 primes.

LINKS

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

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: A143100 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 February 20 06:26 EST 2019. Contains 320332 sequences. (Running on oeis4.)