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
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