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Number of nontrivial divisors of n!.
2

%I #10 Aug 24 2020 23:15:51

%S 0,0,0,2,6,14,28,58,94,158,268,538,790,1582,2590,4030,5374,10750,

%T 14686,29374,41038,60798,95998,191998,242878,340030,532222,677374,

%U 917278,1834558,2332798,4665598,5529598,7864318,12165118,16422910,19595518,39191038,60466174

%N Number of nontrivial divisors of n!.

%C A divisor of n is trivial if it is 1 or n.

%F a(n) = A000005(n!) - 2 for n > 1.

%F a(n) = A070824(n!).

%e The a(3) = 2 through a(5) =14 nontrivial divisions:

%e 6/2 24/2 120/2

%e 6/3 24/3 120/3

%e 24/4 120/4

%e 24/6 120/5

%e 24/8 120/6

%e 24/12 120/8

%e 120/10

%e 120/12

%e 120/15

%e 120/20

%e 120/24

%e 120/30

%e 120/40

%e 120/60

%t Table[Length[DeleteCases[Divisors[n!],1|n!]],{n,10}]

%o (Python)

%o from sympy import factorial, divisor_count

%o def A337106(n):

%o return 0 if n <= 1 else divisor_count(factorial(n))-2 # _Chai Wah Wu_, Aug 24 2020

%Y A070824 counts nontrivial divisors.

%Y A153823 counts proper divisors of n!.

%Y A337107 has this sequence as column k = 3.

%Y A000005 counts divisors.

%Y A000142 lists factorial numbers.

%Y A001055 counts factorizations.

%Y A027423 counts divisors of factorial numbers.

%Y A067824 counts chains of divisors starting with n.

%Y A074206 counts chains of divisors from n to 1.

%Y A076716 counts factorizations of factorial numbers.

%Y A253249 counts chains of divisors.

%Y A337071 counts chains of divisors starting with n!.

%Y A337105 counts chains of divisors from n! to 1.

%Y Cf. A022559, A124010, A251683, A325617, A336941.

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 23 2020

%E a(0) from _Chai Wah Wu_, Aug 24 2020