A058033 Number of powers of 2 between (but not including) two consecutive primorials.

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, 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, 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

Offset: 1

Author

Labos Elemer, Nov 22 2000

Keywords

• nonn

Comments

Does not increase monotonically.

Examples

			Between [2 and 6] is 4;
between [6 and 30] are 8, 16;
between [30 and 210] are 32, 64, 128;
between [210 and 2310] are 256, 512, 1024, 2048;
between [2310 and 30030] are 4096, 8192, 16384;
between [30030, 510510] are 32768, 65536, 131072, 262144.
So a(1), ..., a(6) = 1, 2, 3, 4, 3, 4, ...
From _Michael De Vlieger_, Jul 15 2017: (Start)
First and last positions of values seen in the first 10^5 terms:
Value  First   Last
   1      1       1
   2      2       2
   3      3       5
   4      4       8
   5      7      15
   6     12      27
   7     20      51
   8     35      90
   9     57     161
  10    102     294
  11    182     542
  12    323     995
  13    585    1856
  14   1061    3505
  15   1943    6485
  16   3521   12203
  17   6606   22949
  18  12297   43200
  19  23051   81759
  20  43578  (99999)
  21  82296  (99997)
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A045716, A054850, A084972.

Programs

Mathematica

a = Table[ Floor[ Log[2, Product[ Prime[i], {i, 1, n}]]], {n, 1, 110}]; Drop[a, 1] - Drop[a, -1]
(* Second program: *)
Differences@ Floor@ Log2@ FoldList[Times, Prime@ Range@ 105] (* Michael De Vlieger, Jul 15 2017 *)

Formula

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

First differences of A054850.

Extensions

Edited by Robert G. Wilson v, May 22 2003