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%I #10 Apr 17 2021 01:45:37
%S 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,
%T 5,22,15,47,9,51,17,18,28,19,20,62,12,66,33,68,11,12,38,79,40,27,83,
%U 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
%N Reciprocal of the base-2 logarithm of the ratio between consecutive primes, rounded down.
%C 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.
%F a(n) = floor(1 / log_2(prime(n+1) / prime(n))).
%e 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.
%t f[n_] := Floor[1/Log[2, Prime[n + 1]/Prime[n]]]
%o (PARI) a(n) = log(2)\log(prime(n+1) / prime(n)); \\ _Michel Marcus_, Apr 14 2021
%Y Cf. A000523, A000040, A074461.
%K easy,nonn
%O 1,3
%A _Jack W Grahl_, Jul 07 2007
%E Edited by _Jon E. Schoenfield_, Apr 13 2021