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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  1933  1949  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; count = 0; While[count < max, If[p[] == p[] + 2 && p[] == p[] + 2, d = ((p[] - p[])/2 - 2)/3 + 1; If[d <= max && v[[d]]==0, count++; v[[d]] = p[]]]; p = Join[Rest[p], {NextPrime[p[]]}]]; 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.)