A272863 Numerator of the ratio of consecutive prime gaps.

2, 1, 2, 1, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 1, 3, 2, 1, 3, 2, 3, 4, 1, 1, 2, 1, 2, 7, 2, 3, 1, 5, 1, 3, 1, 2, 3, 1, 1, 5, 1, 2, 1, 6, 1, 1, 1, 2, 3, 1, 5, 3, 1, 1, 1, 3, 2, 1, 5, 7, 2, 1, 2, 7, 3, 5, 1, 2, 3, 4, 3, 1, 2, 3, 4, 1, 2, 5, 1, 5, 1, 3, 2, 3, 4, 1, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

Author

Andres Cicuttin, Jun 21 2016

Keywords

• nonn
• frac

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A001223, A274225 (denominators), A274263.

Cf. A184247 (primes of the form prime(k+1) when a(k)=1).

Programs

Magma

[Numerator((NthPrime(n+2)-NthPrime(n+1))/(NthPrime(n+1)-NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Apr 27 2017

Mathematica

Table[(Prime[j+2]-Prime[j+1])/(Prime[j+1]-Prime[j]),{j,1,120}]//Numerator
Numerator[#[[2]]/#[[1]]]&/@Partition[Differences[Prime[Range[100]]],2,1] (* Harvey P. Dale, Dec 07 2022 *)

PARI

a(n) = numerator((prime(n+2)-prime(n+1))/(prime(n+1)-prime(n))); \\ Michel Marcus, Apr 29 2017

Formula

a(n) = numerator((prime(n+2)-prime(n+1))/(prime(n+1)-prime(n))).

A001223(n) = Product_{k=1..n-1} a(k)/A274225(k). - Andres Cicuttin, Apr 26 2017

A000040(n) = 3 + Sum_{j=1..n-1} Product_{k=1..j} a(k)/A274225(k), for n>1. - Andres Cicuttin, Apr 26 2017