A238716 Run lengths of decadal prime triples.

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, 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, 2, 1, 1

Offset: 1

Author

M. F. Hasler, Mar 03 2014

Keywords

• nonn
• more

Comments

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.

Examples

			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.
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.
The next term A238713(8)=311 starts an "isolated" decadal prime triple, i.e., the next "run" of length a(3)=1.
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).
		

Crossrefs

Cf. A008470, A008471, A238713, A238715.

Programs

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;)}