A359357 Number of different ratios between consecutive prime gaps among the first n primes.

1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 3

Author

Andres Cicuttin, Dec 27 2022

Keywords

• nonn

Comments

What about the asymptotic behavior of a(n), and upper and lower bounds?

Crossrefs

Cf. A001223, A272863, A274225, A274263, A275785.

Programs

Maple

P:= [seq(ithprime(i),i=1..102)]:
G:= P[2..-1]-P[1..-2]:
R:= G[2..-1] /~ G[1..-2]:
seq(nops(convert(R[1..n],set)),n=1..100); # Robert Israel, Jan 25 2023

Mathematica

g[m_]:=(Prime[m+2]-Prime[m+1])/(Prime[m+1]-Prime[m]);
Table[Tally[g[Range[j]]]//Length,{j,1,2^7}]

PARI

f(n) = (prime(n+2)-prime(n+1))/(prime(n+1)-prime(n)); \\ A274263
a(n) = #Set(vector(n, k, f(k))); \\ Michel Marcus, Dec 28 2022

Formula

a(n) = #{A001223(i+1)/A001223(i)}, i < n-1, where A001223 are prime gaps.