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

1,1

Links

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

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

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 *)

Prog

(Magma) [Numerator((NthPrime(n+2)-NthPrime(n+1))/(NthPrime(n+1)-NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Apr 27 2017
(PARI) a(n) = numerator((prime(n+2)-prime(n+1))/(prime(n+1)-prime(n))); \\ Michel Marcus, Apr 29 2017

Crossrefs

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

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

Sequence in context: A103682 A023134 A304536 * A112632 A254575 A355402

Adjacent sequences: A272860 A272861 A272862 * A272864 A272865 A272866

Keyword

nonn,frac

Author

Andres Cicuttin, Jun 21 2016

Status

approved