A142246 Primes of the form (prime(n+2)-prime(n))/(prime(n+1)-prime(n)), where prime(n)=n-th prime.

3, 2, 3, 3, 3, 3, 2, 3, 3, 2, 2, 3, 7, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 2, 2, 7, 3, 2, 7, 2, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 13, 2, 2, 7, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 2, 3, 3, 3, 2, 3, 2, 2, 3, 3, 7, 3, 3, 7, 2, 2, 7, 3, 3, 7, 2, 3, 3, 2, 3, 3, 7

Offset

1,1

Comments

For any prime q, successive prime gaps of 6 and 6*(q-1) will produce an entry of q in this sequence. We therefore expect each prime to occur infinitely often. The first appearances of the first 9 primes are at positions 2, 1, 185, 13, 2091, 60, 20352, 520, and 34248. - Robert Israel, Sep 01 2019

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

If n=1, then (p(1+2)-p(1))/(p(1+1)-p(1))=(5-2)/(3-2)=3=a(1).
If n=2, then (p(2+2)-p(2))/(p(2+1)-p(2))=(7-3)/(5-3)=2=a(2).
If n=3, then (p(3+2)-p(3))/(p(3+1)-p(3))=(11-5)/(7-5)=3= a(3).

Maple

for n from 1 to 1230 do p := (ithprime(n+2)-ithprime(n)) / (ithprime(n+1)-ithprime(n) ) ; if type(p, 'integer') then if isprime(p) then printf("%d, ", p) ; end if; end if; end do: # R. J. Mathar, Apr 28 2010

Mathematica

Select[(#[[3]]-#[[1]])/(#[[2]]-#[[1]])&/@Partition[Prime[ Range[ 1000]], 3, 1], PrimeQ] (* Harvey P. Dale, Aug 20 2017 *)

Crossrefs

Cf. A000040.

Sequence in context: A369100 A060585 A114451 * A080428 A343886 A120004

Adjacent sequences: A142243 A142244 A142245 * A142247 A142248 A142249

Keyword

nonn,less

Author

Juri-Stepan Gerasimov, Oct 06 2008

Extensions

Sequence corrected from a(3) onwards by R. J. Mathar, Apr 28 2010

Status

approved