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Numbers k such that k! has a factorial number of decimal digits.
3

%I #32 Mar 22 2020 10:07:22

%S 0,1,2,3,4,9,24,342,11158,145435633325318659

%N Numbers k such that k! has a factorial number of decimal digits.

%C From _Vaclav Kotesovec_, Mar 22 2020: (Start)

%C a(11) has 1199 digits and a(11)! has 525! digits.

%C a(12) has 1300 digits and a(12)! has 562! digits.

%C a(13) has 3733 digits and a(13)! has 1380! digits.

%C a(14) has 4730 digits and a(14)! has 1693! digits.

%C a(15), if it exists, must have more than 5732 digits and a(15)! must have more than 2000! digits. (End)

%H Vaclav Kotesovec, <a href="/A333431/a333431_1.txt">a(11)-a(14)</a>

%e 9 is in the sequence since 9! = 362880 which has 6 decimal digits and 6 = 3!.

%t f = k = 1; lst = {0}; While[k < 12000, f *= k; If[ MemberQ[{1, 2, 6, 24, 120, 720, 5040, 40320, 362880}, IntegerLength@ f], AppendTo[lst, k]]; k++]; lst

%Y Cf. A000142, A034886.

%K nonn,base

%O 1,3

%A _Robert G. Wilson v_, Mar 20 2020

%E a(10) from _Giovanni Resta_, Mar 21 2020

%E a(11)-a(12) from _Vaclav Kotesovec_, Mar 21 2020

%E a(13)-a(14) from _Vaclav Kotesovec_, Mar 22 2020