

A280201


Let the smallest of three successive primes p, p+d, p+2d be a socalled dtriple and b(n) the sequence of dtriples with d<>6. Then a(n) is the number of 6triples between b(n) and b(n+1).


1



3, 15, 13, 3, 19, 5, 4, 0, 1, 8, 8, 13, 0, 4, 2, 2, 1, 5, 0, 2, 0, 1, 0, 1, 0, 1, 1, 4, 5, 1, 1, 8, 3, 1, 1, 3, 3, 2, 4, 2, 2, 2, 0, 1, 2, 5, 1, 1, 2, 2
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OFFSET

1,1


COMMENTS

The sequence of all dtriples A122535(n) = (3), 47, 151, 167, (199), 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, (1499), ... is the union of A047948(n) with 6triples and b(n) with terms in brackets. There are three 6triples between 3 and 199 and 15 6triples between 199 and 1499. Thus a(1)=3 (see example) and a(2)=15.
The average of the first 10 terms is (3+15+13+3+19+5+4+0+1+8)/10 = 7.1. This means that, in this section, the 6triples are more than 7 times as frequent as the other dtriples as a whole. Let us compare longer sections of a(n) with different magnitudes of n, for example (with S(n)=sum(a(k),k,1,n)/n): n <= 10000 100000 733158
S(n) = 1.28 0.98 0.81
n=733158 was the largest available index when I analyzed a pool of primes <=10^9.
Result: For small n, 6triples are more frequent than the whole of other dtriples; for large n, the reverse is true. Does S(n) tend to zero? It seems so, see link "Tendency of a(n)".  Gerhard Kirchner, Dec 28 2016


LINKS

Gerhard Kirchner, Table of n, a(n) for n = 1..10000
Gerhard Kirchner, Tendency of a(n)


EXAMPLE

The first dtriples are 3 (,5,7, d=2); 47 (,53,59, d=6); 151 (,157,163, d=6); 167 (,173,179, d=6); 199 (,211,223, d=12). So there are three 6triples between the 2triple and the 12triple: a(1)=3.


CROSSREFS

Cf. A047948, A122535.
Sequence in context: A232076 A099476 A063628 * A296842 A279925 A279534
Adjacent sequences: A280198 A280199 A280200 * A280202 A280203 A280204


KEYWORD

nonn


AUTHOR

Gerhard Kirchner, Dec 28 2016


STATUS

approved



