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A085301
Number of factorials between two primorials.
4
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
1,1
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.
LINKS
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
EXAMPLE
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.
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 *)
KEYWORD
nonn,changed
AUTHOR
Labos Elemer, Jun 26 2003
STATUS
approved