A130742 - OEIS

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

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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.