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A107926 The least number k such that there are primes p and q with p - q = 2*n, p + q = k, and p the least such prime >= k/2. 6
4, 8, 18, 16, 54, 48, 50, 108, 102, 44, 234, 444, 98, 228, 174, 92, 414, 432, 242, 516, 582, 256, 1182, 672, 406, 612, 846, 272, 1038, 984, 442, 1776, 1902, 292, 1074, 636, 1054, 3312, 1122, 476, 1398, 1464, 530, 1728, 2730, 572, 2706, 3348, 682, 2844, 3342 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

From the Goldbach conjecture.

A107926 = 2*A103147 by definition.

a(3n)> a(3n-2), a(3n-1), a(3n+1) & a(3n+2) for all n > 0 except for n = 1, 2, 12, 19, 20 or 41.

Of those values found so far a(3n+2) > a(3n+1) by ~8%. - Robert G. Wilson v, Nov 03 2013

Except for 1, all indices, i, not congruent to 0 (mod 3), a(i) is congruent to 0 (mod 6) and for all indices, i, congruent to 0 (mod 3), a(i) is not congruent to 0 (mod 6). Of those not congruent to 0 (mod 6), those congruent to 2 outnumber those congruent to 4, about 8 to 7. Robert G. Wilson v, Nov 03 2013

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..2560 (first 501 terms from T. D. Noe)

Mark Herkommer, Goldbach Conjecture Research.

Tomás Oliveira e Silva, Goldbach conjecture verification.

The Prime Glossary, Goldbach's conjecture

+Plus Magazine ... living mathematics, Mathematical mysteries: the Goldbach conjecture

Eric Weisstein's World of Mathematics, Goldbach Conjecture

Wikipedia, Goldbach conjecture

EXAMPLE

a(0) = 4 because 4=2+2 and 2-2=0.

a(1) = 8 because 8 is the least number with 8=p+q and p-q=2 for primes p and q.

a(2) = 18 because 18=7+11 and the primes 7 and 11 have difference 4.

MATHEMATICA

f[n_] := For[p = n/2, True, p--, If[PrimeQ[p] && PrimeQ[n - p], Return[n/2 - p]]]; nn=101; t=Table[0, {nn}]; cnt=0; n=1; While[cnt<nn, n++; d=f[2n]; If[d+1<=nn && t[[d+1]]==0, t[[d+1]]=n; cnt++]]; 2t

CROSSREFS

Cf. A066285, A103147, records in A065978 and A066286.

Sequence in context: A312825 A312826 A110601 * A174741 A312827 A308150

Adjacent sequences:  A107923 A107924 A107925 * A107927 A107928 A107929

KEYWORD

nonn

AUTHOR

Gilmar J. Rodriguez (Gilmar.Rodriguez(AT)nwfwmd.state.fl.us) and Robert G. Wilson v, Jun 16 2005

STATUS

approved

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)