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

%I A058033 #15 Jul 17 2017 02:34:02
%S A058033 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 A058033 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 A058033 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 A058033 Number of powers of 2 between (but not including) two consecutive primorials.
%C A058033 Does not increase monotonically.
%H A058033 Michael De Vlieger, <a href="/A058033/b058033.txt">Table of n, a(n) for n = 1..10000</a>
%F A058033 a(n)= Cardinality{b |A002110(n)<= b < A002110(n+1), b=2^x for some x} a(n)=A045716(n+1)-A045716(n)
%F A058033 First differences of A054850.
%e A058033 Between [2 and 6] is 4;
%e A058033 between [6 and 30] are 8, 16;
%e A058033 between [30 and 210] are 32, 64, 128;
%e A058033 between [210 and 2310] are 256, 512, 1024, 2048;
%e A058033 between [2310 and 30030] are 4096, 8192, 16384;
%e A058033 between [30030, 510510] are 32768, 65536, 131072, 262144.
%e A058033 So a(1), ..., a(6) = 1, 2, 3, 4, 3, 4, ...
%e A058033 From _Michael De Vlieger_, Jul 15 2017: (Start)
%e A058033 First and last positions of values seen in the first 10^5 terms:
%e A058033 Value  First   Last
%e A058033    1      1       1
%e A058033    2      2       2
%e A058033    3      3       5
%e A058033    4      4       8
%e A058033    5      7      15
%e A058033    6     12      27
%e A058033    7     20      51
%e A058033    8     35      90
%e A058033    9     57     161
%e A058033   10    102     294
%e A058033   11    182     542
%e A058033   12    323     995
%e A058033   13    585    1856
%e A058033   14   1061    3505
%e A058033   15   1943    6485
%e A058033   16   3521   12203
%e A058033   17   6606   22949
%e A058033   18  12297   43200
%e A058033   19  23051   81759
%e A058033   20  43578  (99999)
%e A058033   21  82296  (99997)
%e A058033 (End)
%t A058033 a = Table[ Floor[ Log[2, Product[ Prime[i], {i, 1, n}]]], {n, 1, 110}]; Drop[a, 1] - Drop[a, -1]
%t A058033 (* Second program: *)
%t A058033 Differences@ Floor@ Log2@ FoldList[Times, Prime@ Range@ 105] (* _Michael De Vlieger_, Jul 15 2017 *)
%Y A058033 Cf. A002110, A045716, A054850, A084972.
%K A058033 nonn
%O A058033 1,2
%A A058033 _Labos Elemer_, Nov 22 2000
%E A058033 Edited by _Robert G. Wilson v_, May 22 2003