A085301 Number of factorials between two primorials.

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

Author

Labos Elemer, Jun 26 2003

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

Formula

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