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Number of proper divisors of n!.
3

%I #19 Sep 08 2022 08:45:40

%S 0,0,1,3,7,15,29,59,95,159,269,539,791,1583,2591,4031,5375,10751,

%T 14687,29375,41039,60799,95999,191999,242879,340031,532223,677375,

%U 917279,1834559,2332799,4665599,5529599,7864319,12165119,16422911

%N Number of proper divisors of n!.

%C a(n) is the number of proper divisors of factorial number A000142(n).

%H G. C. Greubel, <a href="/A153823/b153823.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = A000005(A000142(n)) - 1 = A032741(A000142(n)) = A027423(n) - 1.

%e For n=4, 4! = 4*3*2*1 = 24, which has 7 proper divisors: 1, 2, 3, 4, 6, 8, and 12. So a(4) = 7. - _Michael B. Porter_, Aug 30 2016

%t Table[DivisorSigma[0, n!] - 1, {n, 0, 50}] (* _G. C. Greubel_, Aug 30 2016 *)

%o (Magma) [DivisorSigma(0,Factorial(n)) - 1: n in [0..40]]; // _Vincenzo Librandi_, Aug 31 2016

%o (PARI) a(n) = numdiv(n!) - 1; \\ _Michel Marcus_, Aug 31 2016

%o (Python)

%o from sympy import factorial, divisor_count

%o def A153823(n):

%o return divisor_count(factorial(n))-1 # _Chai Wah Wu_, Aug 24 2020

%Y Cf. A000005, A000142, A027423, A032741.

%Y Cf. A153823, A153825. [From _Omar E. Pol_, Jan 17 2009]

%K nonn

%O 0,4

%A _Omar E. Pol_, Jan 02 2009

%E More terms from _Omar E. Pol_, Jan 17 2009