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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

%I #113 Apr 03 2023 10:36:10

%S 4,8,18,16,54,48,50,108,102,44,234,444,98,228,174,92,414,432,242,516,

%T 582,256,1182,672,406,612,846,272,1038,984,442,1776,1902,292,1074,636,

%U 1054,3312,1122,476,1398,1464,530,1728,2730,572,2706,3348,682,2844,3342

%N 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.

%C From the Goldbach conjecture.

%C A107926 = 2*A103147 by definition.

%C 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.

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

%C 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

%H Robert G. Wilson v, <a href="/A107926/b107926.txt">Table of n, a(n) for n = 0..2560</a> (first 501 terms from T. D. Noe)

%H Mark Herkommer, <a href="http://www.petrospec-technologies.com/Herkommer/goldbach.htm">Goldbach Conjecture Research</a>.

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>.

%H The Prime Glossary, <a href="https://t5k.org/glossary/xpage/goldbachconjecture.html">Goldbach's conjecture</a>

%H +Plus Magazine ... living mathematics, <a href="http://plus.maths.org/content/mathematical-mysteries-goldbach-conjecture">Mathematical mysteries: the Goldbach conjecture</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach&#39;s_conjecture">Goldbach conjecture</a>

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

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

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

%t 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

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

%K nonn

%O 0,1

%A Gilmar J. Rodriguez (Gilmar.Rodriguez(AT)nwfwmd.state.fl.us) and _Robert G. Wilson v_, Jun 16 2005