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A082512 a(n) = p is the smallest prime introducing a consecutive prime-difference 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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 21 12:13 EST 2022. Contains 350477 sequences. (Running on oeis4.)