A084972 -id:A084972 - cp's OEIS frontend

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

Labos Elemer, Nov 22 2000

nonn

Does not increase monotonically.

			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)

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

Cf. A002110, A045716, A054850, A084972.

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 *)

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.

Edited by Robert G. Wilson v, May 22 2003

A085301 Number of factorials between two primorials.

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 1

Labos Elemer, Jun 26 2003

nonn

Seems provable: a(n) > 0 for all n; seems more difficult to prove (if true at all) that a(n)=1 or 2; for n < 2050 it holds. Stirling's approximation and Prime Number Theorem together may help.

			n=1: between 1st (=2) and 2nd (=6) primorials, the factorials 2!=2 and 3!=6 occur, so a(1)=2.
n=2: between the primorials 6 and 30, the factorials 3!=6 and 4!=24 occur, so a(2)=2.
Factorial and primorial sets coincide only in case of n = 1,2: {2,6}.
If n > 3, factorials are never squarefree; but primorials are always squarefree, so they are disjoint.
n=5: between the 5th and 6th primorials 2310 and 30030, only the factorial 7!=5040 occurs.
n=6: between the primorials 30030 and 510510, the factorials 8!=40320 and 9!=362880 occur.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A002110, A067850, A084320, A084321, A084558, A084972, A085355.

fn[n_] := Module[{k = 1, r = n}, While[r >= 1, k++; r /= k]; k - 1];
prim[n_] := Times @@ Prime[Range[n]];
a[n_] := fn[prim[n]] - fn[prim[n - 1]]; a[1] = a[2] = 2; Array[a, 100] (* Amiram Eldar, Oct 24 2024 *)

a(n) = Card[{k; q(n) <= k! <= q(n+1)}, where q(j) = A002110(j), the j-th primorial; closed intervals required only for n = 1, 2.

a(n) = A084558(A002110(n)) - A084558(A002110(n-1)) for n >= 3. - Amiram Eldar, Oct 24 2024

cp's OEIS Frontend

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

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