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Number of powers of 2 between (but not including) two consecutive primorials.
5

%I #15 Jul 17 2017 02:34:02

%S 1,2,3,4,3,4,5,4,5,5,5,6,5,6,5,6,6,6,6,7,6,6,7,6,7,7,6,7,7,7,7,7,7,7,

%T 8,7,7,8,7,8,7,8,7,8,8,7,8,8,8,8,7,8,8,8,8,8,9,8,8,8,8,8,9,8,8,9,8,9,

%U 8,8,9,8,9,9,8,9,8,9,9,8,9,9,9,8,9,9,9,9,9,8,9,9,9,9,9,9,9,9,9,9,9,10,9,9,9

%N Number of powers of 2 between (but not including) two consecutive primorials.

%C Does not increase monotonically.

%H Michael De Vlieger, <a href="/A058033/b058033.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n)= Cardinality{b |A002110(n)<= b < A002110(n+1), b=2^x for some x} a(n)=A045716(n+1)-A045716(n)

%F First differences of A054850.

%e Between [2 and 6] is 4;

%e between [6 and 30] are 8, 16;

%e between [30 and 210] are 32, 64, 128;

%e between [210 and 2310] are 256, 512, 1024, 2048;

%e between [2310 and 30030] are 4096, 8192, 16384;

%e between [30030, 510510] are 32768, 65536, 131072, 262144.

%e So a(1), ..., a(6) = 1, 2, 3, 4, 3, 4, ...

%e From _Michael De Vlieger_, Jul 15 2017: (Start)

%e First and last positions of values seen in the first 10^5 terms:

%e Value First Last

%e 1 1 1

%e 2 2 2

%e 3 3 5

%e 4 4 8

%e 5 7 15

%e 6 12 27

%e 7 20 51

%e 8 35 90

%e 9 57 161

%e 10 102 294

%e 11 182 542

%e 12 323 995

%e 13 585 1856

%e 14 1061 3505

%e 15 1943 6485

%e 16 3521 12203

%e 17 6606 22949

%e 18 12297 43200

%e 19 23051 81759

%e 20 43578 (99999)

%e 21 82296 (99997)

%e (End)

%t a = Table[ Floor[ Log[2, Product[ Prime[i], {i, 1, n}]]], {n, 1, 110}]; Drop[a, 1] - Drop[a, -1]

%t (* Second program: *)

%t Differences@ Floor@ Log2@ FoldList[Times, Prime@ Range@ 105] (* _Michael De Vlieger_, Jul 15 2017 *)

%Y Cf. A002110, A045716, A054850, A084972.

%K nonn

%O 1,2

%A _Labos Elemer_, Nov 22 2000

%E Edited by _Robert G. Wilson v_, May 22 2003