A276309 Integer part of the ratio of alternate consecutive prime gaps.

2, 2, 1, 1, 1, 1, 3, 0, 1, 2, 0, 1, 3, 1, 0, 1, 2, 0, 1, 2, 1, 2, 0, 0, 1, 1, 1, 7, 1, 0, 0, 1, 1, 0, 3, 0, 1, 1, 0, 1, 1, 0, 1, 3, 6, 0, 0, 1, 3, 0, 1, 3, 0, 1, 0, 1, 2, 0, 2, 7, 0, 0, 1, 7, 1, 0, 0, 0, 3, 2, 1, 0, 0, 1, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 2, 0, 0, 1, 6, 2, 0, 1, 1, 0, 3, 0

Offset: 1

Author

Andres Cicuttin, Aug 06 2016

Keywords

• nonn

Comments

Conjectures: The most frequent ratio among alternate prime gaps is 1, while the most frequent ratios among consecutive prime gaps seems to be 2 and 1/2, both with nearly the same frequency (see links). It also appears that next four most frequent ratios are 2, 1/2, 3 and 1/3, all four with nearly the same frequency (see links).

Examples

			For n=2, the second prime is 3, and the next three primes are 5, 7, and 11. So the ratio of prime gaps is (11-7)/(5-3) = 4/2 = 2, and the integer part of this is a(2) = 2. - _Michael B. Porter_, Aug 11 2016
		

Links

• Andres Cicuttin, Counts on first 200000 ratios of alternate prime gaps
• Andres Cicuttin, Several histograms of the logarithm of the ratio of consecutive prime gaps (prime(n+2)-prime(n+1))/(prime(n+1)-prime(n)) obtained for the first 2*10^5 primes with different bin sizes

Crossrefs

Cf. A001223, A274263, A272863, A274225.

Programs

Magma

[Floor((NthPrime(n+3)-NthPrime(n+2))/(NthPrime(n+ 1)- NthPrime(n))): n in [1..100]]; // Vincenzo Librandi, Aug 30 2016

Mathematica

Table[Floor[(Prime[j + 3] - Prime[j + 2])/(Prime[j + 1] - Prime[j])], {j, 1, 200}]

Formula

a(n) = floor((prime(n + 3) - prime(n + 2))/(prime(n + 1) - prime(n))) .