A153823 -id:A153823 - cp's OEIS frontend

A153825 Sum of proper divisors minus the number of proper divisors of n!.

0, 0, 0, 3, 29, 225, 1669, 14245, 118705, 1118001, 11705019, 144091717, 1738439017, 24817157329, 355309322689, 5378578597729, 86081749391905, 1570394279028289, 28281459220178401, 572031558109560385, 11458497230555053681

Offset: 0

Omar E. Pol, Jan 02 2009

nonn

a(n) is the sum of proper divisors minus the number of proper divisors of factorial number A000142(n).

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000005, A000142, A000203, A001065, A032741, A062569, A152770, A153823, A153824.

[DivisorSigma(1,Factorial(n)) - Factorial(n) - (DivisorSigma(0,Factorial(n))-1): n in [0..20]]; // Vincenzo Librandi, Aug 31 2016

with(numtheory): seq(sigma(factorial(n))-factorial(n)-tau(factorial(n))+1, n = 0 .. 22); # Emeric Deutsch, Jan 07 2009

Table[DivisorSigma[1,n!]-n!-(DivisorSigma[0,n!]-1),{n,0,20}] (* Harvey P. Dale, Jan 14 2012 *)

a(n) = (sigma(n!) - n!) - (numdiv(n!) - 1); \\ Michel Marcus, Aug 31 2016

a(n) = A153824(n) - A153823(n) = A152770(A000142(n)).

Extended by Emeric Deutsch, Jan 07 2009

A337106 Number of nontrivial divisors of n!.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 28, 58, 94, 158, 268, 538, 790, 1582, 2590, 4030, 5374, 10750, 14686, 29374, 41038, 60798, 95998, 191998, 242878, 340030, 532222, 677374, 917278, 1834558, 2332798, 4665598, 5529598, 7864318, 12165118, 16422910, 19595518, 39191038, 60466174

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

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

			The a(3) = 2 through a(5) =14 nontrivial divisions:
  6/2  24/2   120/2
  6/3  24/3   120/3
       24/4   120/4
       24/6   120/5
       24/8   120/6
       24/12  120/8
              120/10
              120/12
              120/15
              120/20
              120/24
              120/30
              120/40
              120/60

A070824 counts nontrivial divisors.
A153823 counts proper divisors of n!.
A337107 has this sequence as column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains of divisors starting with n!.
A337105 counts chains of divisors from n! to 1.
Cf. A022559, A124010, A251683, A325617, A336941.

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

from sympy import factorial, divisor_count
def A337106(n):
    return 0 if n <= 1 else divisor_count(factorial(n))-2 # Chai Wah Wu, Aug 24 2020

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

a(n) = A070824(n!).

a(0) from Chai Wah Wu, Aug 24 2020

cp's OEIS Frontend

A153825 Sum of proper divisors minus the number of proper divisors of n!.

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