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A238716 Run lengths of decadal prime triples.

%I A238716 #16 Feb 11 2023 08:07:27
%S A238716 5,2,1,2,2,3,3,1,3,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,1,
%T A238716 1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,3,
%U A238716 2,1,1
%N A238716 Run lengths of decadal prime triples.
%C A238716 Length of runs of "consecutive" (step = 3) values in A008470, which 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.
%e A238716 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. Therefore, a(1)=5.
%e A238716 The next "decadal prime triples" start at A238713(6)=191 and A238713(7)=223, they form the next run of length a(2)=2, since the decades A008470(6)=19 and A008470(7)=22 differ by the minimum which is 3, but the next one is further away.
%e A238716 The next term A238713(8)=311 starts an "isolated" decadal prime triple, i.e., the next "run" of length a(3)=1.
%e A238716 The next run of length 4 starts with decade m=541, and the next occurrence of 5 consecutive triples starts with decade m=910052463685 (found by J. K. Andersen).
%o A238716 (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; print1(i",");i>=3 && print1("/*",[nextprime(d-10-30*i),precprime(d-30)]"*/ ");i=0;)}
%Y A238716 Cf. A008470, A008471, A238713, A238715.
%K A238716 nonn,more
%O A238716 1,1
%A A238716 _M. F. Hasler_, Mar 03 2014