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

1,3

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.

Links

Table of n, a(n) for n=1..77.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A000523, A000040, A074461.

Sequence in context: A225368 A099311 A349442 * A130107 A107130 A194747

Adjacent sequences: A130739 A130740 A130741 * A130743 A130744 A130745

Keyword

easy,nonn

Author

Jack W Grahl, Jul 07 2007

Extensions

Edited by Jon E. Schoenfield, Apr 13 2021

Status

approved