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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 (list; graph; refs; listen; history; text; internal format)
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 b-files I have extended the current b-file 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+d-1], 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

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Last modified June 16 14:16 EDT 2021. Contains 345057 sequences. (Running on oeis4.)