A086140 - OEIS

%I #28 Apr 11 2022 22:21:26

%S 5,7,11,97,101,1481,1867,3457,5647,15727,16057,16061,19417,19421,

%T 21011,22271,43777,43781,55331,79687,88807,101107,144161,165701,

%U 166841,195731,201821,225341,247601,257857,266677,268811,276037,284737,326141,340927

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

%C p+12 must be a prime. - _Harvey P. Dale_, Jun 11 2015

%C 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 5-tuples 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

%H Harvey P. Dale and Gerhard Kirchner, <a href="/A086140/b086140.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)

%H Gerhard Kirchner, <a href="/A086140/a086140.pdf">Comparison with an assumed asymptotic distribution</a>

%e There are two types of prime 5-tuples, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime 5-tuple 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 5-tuple of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence. - _Michael B. Porter_, Dec 19 2016

%t 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}]

%t (* Second program: *)

%t Transpose[Select[Partition[Prime[Range[30000]],5,1],#[[5]]-#[[1]] == 12&]][[1]] (* _Harvey P. Dale_, Jun 11 2015 *)

%Y Cf. A031930, A046133, A086139, A086136, A022006, A022007, A001359 (twins), A007529 (triples), A007530 (quadruplets).

%K nonn

%O 1,1

%A _Labos Elemer_, Jul 29 2003