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.

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

Offset: 0

Author

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

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

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

Programs

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