A238715 Least prime of a run of 3 or more consecutive decadal prime triples.

11, 821, 1031, 1423, 5413, 13691, 140831, 220873, 266023, 283571, 464741, 1596311, 1660661, 1966813, 2655403, 3303341, 5191331, 5485393, 8125511, 14241911, 14848511, 15586993, 15852043, 16539163, 19608041, 19696841, 30624071, 31809073, 35493551, 38335541, 40430771

Offset

1,1

Comments

Sequence A008470 lists "prime triple decades", i.e., numbers m>1 such that the interval (10m,10m+10) contains at least 3 primes. The decades must be of the form m=3k+1, since for m=3k, 10m+3 and 10m+9 cannot be prime and for m=3k+2, 10k+1 and 10k+7 cannot be prime. Thus, "consecutive" prime triples are meant here in the sense of consecutive k-values.

Alternatively, the present sequence lists the terms A238713(n) for which A238713(n+2) <= A238713(n)+75, or equivalently, floor(A238713(n+2)/30) <= floor(A238713(n)/30)+2, but only if A238713(n-1) < A238713(n)-15, to keep only the first of a possibly longer run, cf. example.

See A238716 for the length of the runs of "consecutive" decades A008470 in this sense.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Example

The first occurrence of 5 consecutive triples is: {11, 13, 17 (or 19)} ; {41, 43, 47} ; {71, 73, 79} ; {101, 103, 107 (or 109)} ; {131, 137, 139}. This corresponds to decades 1,4,7,10,13; i.e., the first 5 terms of sequence A008470. The present sequence only lists a(1)=11, but not 41 or 71 which also start a run of 3 consecutive prime triple decades, but they are not listed because already part of the run starting at a(1).
The next occurrence of 4 consecutive triples starts with decade m=541, and the next occurrence of 5 consecutive triples starts with decade m=910052463685, at p = 9100524636851 (found by J. K. Andersen).

Prog

(PARI) {d=10; p=primepi(d); i=0; while( po=p, p=primepi( d+=10 ); p>2+po && i++ && (p=primepi(d+=20)) && next; i || next; i>=3 && print1(nextprime(d-10-30*i)", "); i=0; )} \\ this could be optimized ...
(PARI) isA238713(n)=my(t=n%10); if(t==1, isprime(n) && if(isprime(n+2), isprime(n+6) || isprime(n+8), isprime(n+6) && isprime(n+8)), t==3 && isprime(n) && !isprime(n-2) && isprime(n+4) && isprime(n+6))
isA008470(n)=if(isprime(10*n+1), if(isprime(10*n+3), isprime(10*n+7) || isprime(10*n+9), isprime(10*n+7) && isprime(10*n+9)), isprime(10*n+3) && isprime(10*n+7) && isprime(10*n+9))
is(n)=isA238713(n) && isA008470(n\10+3) && isA008470(n\10+6) && !isA008470(n\10-3) \\ Charles R Greathouse IV, Mar 04 2014

Crossrefs

Cf. A008470, A008471, A238713, A238730.

Sequence in context: A100369 A196426 A371720 * A162448 A024150 A052071

Adjacent sequences: A238712 A238713 A238714 * A238716 A238717 A238718

Keyword

nonn

Author

M. F. Hasler, Mar 03 2014

Extensions

a(20)-a(31) from Charles R Greathouse IV, Mar 04 2014

Status

approved