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%I #26 Jan 19 2019 04:15:43
%S 11,821,1031,1423,5413,13691,140831,220873,266023,283571,464741,
%T 1596311,1660661,1966813,2655403,3303341,5191331,5485393,8125511,
%U 14241911,14848511,15586993,15852043,16539163,19608041,19696841,30624071,31809073,35493551,38335541,40430771
%N Least prime of a run of 3 or more consecutive decadal prime triples.
%C 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.
%C 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.
%C See A238716 for the length of the runs of "consecutive" decades A008470 in this sense.
%H Charles R Greathouse IV, <a href="/A238715/b238715.txt">Table of n, a(n) for n = 1..500</a>
%e 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).
%e 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).
%o (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 ...
%o (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))
%o 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))
%o is(n)=isA238713(n) && isA008470(n\10+3) && isA008470(n\10+6) && !isA008470(n\10-3) \\ _Charles R Greathouse IV_, Mar 04 2014
%Y Cf. A008470, A008471, A238713, A238730.
%K nonn
%O 1,1
%A _M. F. Hasler_, Mar 03 2014
%E a(20)-a(31) from _Charles R Greathouse IV_, Mar 04 2014
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