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%I #9 Sep 05 2017 03:13:22
%S 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,
%T 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,
%U 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
%N Number of factorials between two primorials.
%C 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.
%F 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.
%e n=1: between 1st (=2) and 2nd (=6) primorials, the factorials 2!=2 and 3!=6 occur, so a(1)=2;
%e n=2: between the primorials 6 and 30, the factorials 3!=6 and 4!=24 occur, so a(2)=2.
%e Factorial and primorial sets coincide only in case of n = 1,2: {2,6}.
%e If n > 3, factorials are never squarefree; but primorials are always squarefree, so they are disjoint.
%e n=5: between the 5th and 6th primorials 2310 and 30030, only the factorial 7!=5040 occurs;
%e n=6: between the primorials 30030 and 510510, the factorials 8!=40320 and 9!=362880 occur.
%Y Cf. A000142, A002110, A067850, A084320, A084321, A084972, A085355.
%K nonn
%O 1,1
%A _Labos Elemer_, Jun 26 2003
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