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Number of divisors of (n!)^n (A036740).
0

%I #12 Jan 04 2024 21:58:50

%S 1,1,3,16,65,576,2275,27840,78489,236800,767151,13264560,31184725,

%T 640564848,2082421125,5514535936,10924376001,279876280320,

%U 584912713825,16888996800000,37538691697521,91766133606400,272224952406045,9248286151802880,17279191734765625

%N Number of divisors of (n!)^n (A036740).

%C Divisible by n+1. Proof: Exponent of largest prime dividing n! in prime factorization of n! is 1, i.e., n! = p_1^e_1*p_2^e_2*...*p_(s-1)^e_(s-1)*p_s, p_1<p_2<....<p_(s-1)<p_s. Thus tau(n!^n) = (n*e_1+1)*(n*e_2+1)*...*(n*e_(s-1)+1)*(n+1). - _Vladeta Jovovic_, Oct 01 2004

%F a(n) = A000005(A036740(n)).

%o (PARI) for(n=0,22,print(numdiv((n!)^n)))

%Y Cf. A000005, A027423, A036740.

%K easy,nonn

%O 0,3

%A _Jason Earls_, Jul 22 2001