

A085301


Number of factorials between two primorials.


3



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
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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

Table of n, a(n) for n=1..105.


FORMULA

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


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.


CROSSREFS

Cf. A000142, A002110, A067850, A084320, A084321, A084972, A085355.
Sequence in context: A037803 A184318 A030410 * A138385 A030614 A128016
Adjacent sequences: A085298 A085299 A085300 * A085302 A085303 A085304


KEYWORD

nonn


AUTHOR

Labos Elemer, Jun 26 2003


STATUS

approved



