A130742 Reciprocal of the base-2 logarithm of the ratio between consecutive primes, rounded down.

1, 1, 2, 1, 4, 2, 6, 3, 2, 10, 3, 6, 14, 7, 5, 6, 20, 7, 11, 24, 8, 14, 9, 8, 17, 35, 18, 37, 19, 5, 22, 15, 47, 9, 51, 17, 18, 28, 19, 20, 62, 12, 66, 33, 68, 11, 12, 38, 79, 40, 27, 83, 17, 29, 30, 30, 93, 31, 48, 97, 19, 14, 53, 108, 54, 16, 38, 23, 120, 60, 41, 31, 42, 43, 66, 44, 34

Offset: 1

Author

Jack W Grahl, Jul 07 2007

Keywords

• easy
• nonn

Comments

a(n) is the largest power to which the fraction prime(n+1)/prime(n) can be raised without yielding a result which is greater than 2. It has been proved that lim inf of this sequence is (positive) infinity; e.g., the ratio between subsequent primes tends to 1.

Examples

			a(5) = 4 because the sixth prime, 13, divided by the fifth prime, 11, has base-two logarithm 0.241008... and this lies between 1/4 and 1/5.
		

Crossrefs

Cf. A000523, A000040, A074461.

Programs

Mathematica

f[n_] := Floor[1/Log[2, Prime[n + 1]/Prime[n]]]

PARI

a(n) = log(2)\log(prime(n+1) / prime(n)); \\ Michel Marcus, Apr 14 2021

Formula

a(n) = floor(1 / log_2(prime(n+1) / prime(n))).

Extensions

Edited by Jon E. Schoenfield, Apr 13 2021