

A082512


a(n) = p is the smallest prime introducing a consecutive primedifference pattern as follows: [2,2n,2], i.e., [p, p+2, p+2+2n, p+2+2n+2] are consecutive primes. Increasing middle prime gap in the immediate neighborhood of two small gaps(=2); a(n) = 0 if no such pattern exists.


1



0, 5, 0, 0, 137, 0, 0, 1931, 0, 0, 9437, 0, 0, 2969, 0, 0, 20441, 0, 0, 62987, 0, 0, 510401, 0, 0, 48677, 0, 0, 677471, 0, 0, 997811, 0, 0, 173357, 0, 0, 1134311, 0, 0, 3063287, 0, 0, 3591191, 0, 0, 4876511, 0, 0, 838247, 0, 0, 4297091, 0, 0, 15492437, 0, 0, 27458747
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OFFSET

1,2


COMMENTS

It is conjectured that the twin primes in the neighborhood can be separated by an arbitrarily large gap.


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..225


FORMULA

a(n) = 0 if n != 2 (mod 3).  Amiram Eldar, Jan 21 2020


EXAMPLE

a(4) = 0 because no p can begin a [2,8,2] gap pattern since p mod 6 = 5 must hold and following 3 primes give modulo 6 residues 1, 3, and 5, so p + 2 + 8 is not prime; a(n)=0 if 2n congruent to 0 or 2 mod 6; a(n) has solution for n = 6k + 4;
For n=16, the 4 corresponding primes and 3 differences are {1931 [2] 1933 [16] 1949 [2] 1951}.


MATHEMATICA

d[x_] := Prime[x+1]Prime[x]; h={k1=2, k2=82, k3=2}; de=Apply[Plus, h]; k=0; Do[If[Equal[d[n], k1]&&Equal[d[n+1], k2]&&Equal[d[n+2], k3], k=k+1; Print[k, n, h, {Prime[n], Prime[n+1], Prime[n+2], Prime[n+3]}]], {n, 1, 10000000}]
max = 20; v = Table[0, {max}]; p = Prime /@ Range[4]; count = 0; While[count < max, If[p[[2]] == p[[1]] + 2 && p[[4]] == p[[3]] + 2, d = ((p[[3]]  p[[2]])/2  2)/3 + 1; If[d <= max && v[[d]]==0, count++; v[[d]] = p[[1]]]]; p = Join[Rest[p], {NextPrime[p[[4]]]}]]; Riffle[Table[0, {2*max}], v, {2, 1, 3}] (* Amiram Eldar, Jan 21 2020 *)


CROSSREFS

Sequence in context: A083527 A221240 A113038 * A068385 A318657 A286277
Adjacent sequences: A082509 A082510 A082511 * A082513 A082514 A082515


KEYWORD

nonn


AUTHOR

Labos Elemer, Apr 29 2003


EXTENSIONS

Corrected by T. D. Noe, Nov 15 2006
a(50) corrected and more terms added by Amiram Eldar, Jan 21 2020


STATUS

approved



