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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 (list; graph; refs; listen; history; text; internal format)
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 j-th 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

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Last modified July 20 15:59 EDT 2019. Contains 325185 sequences. (Running on oeis4.)