

A086140


Primes p such that three (the maximum number) primes occur between p and p+12.


29



5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, 16061, 19417, 19421, 21011, 22271, 43777, 43781, 55331, 79687, 88807, 101107, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 257857, 266677, 268811, 276037, 284737, 326141, 340927
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OFFSET

1,1


COMMENTS

p+12 must be a prime.  Harvey P. Dale, Jun 11 2015
A086140 is the union of A022006 and A022007. By merging the two bfiles I have extended the current bfile up to n=10000 (nearly n=20000 would have been possible). I add a comparison (see Links) between the frequency of prime quintuplets and an asymptotic approximation, which is unproven but likely to be true, and based on a conjecture first published by Hardy and Littlewood in 1923. Twins, triples and quadruplets are treated as well.  Gerhard Kirchner, Dec 07 2016


LINKS

Harvey P. Dale and Gerhard Kirchner, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
Gerhard Kirchner, Comparison with an assumed asymptotic distribution


EXAMPLE

There are two types of prime quintuplets, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime quintuplet of the form (p, p+2, p+6, p+8, p+12), so 11 is in the sequence, and (97, 101, 103, 107, 109) is a prime quintuplet of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence.  Michael B. Porter, Dec 19 2016


MATHEMATICA

cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=12, k=0}; Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&&Equal[cp[s+1, s+d1], 3], k=k+1; Print[s]], {n, 1, 100000}]
(* Second program: *)
Transpose[Select[Partition[Prime[Range[30000]], 5, 1], #[[5]]#[[1]] == 12&]][[1]] (* Harvey P. Dale, Jun 11 2015 *)


CROSSREFS

Cf. A031930, A046133, A086139, A086136, A022006, A022007, A001359 (twins), A007529 (triples), A007530 (quadruplets).
Sequence in context: A091509 A027728 A218275 * A104387 A133761 A057659
Adjacent sequences: A086137 A086138 A086139 * A086141 A086142 A086143


KEYWORD

nonn


AUTHOR

Labos Elemer, Jul 29 2003


STATUS

approved



