A337105 -id:A337105 - cp's OEIS frontend

A337071 Number of strict chains of divisors starting with n!.

1, 1, 2, 6, 40, 264, 3776, 40256, 1168000, 34204032, 1107791872, 23233380352, 1486675898368, 38934372315136, 1999103691427840, 132874800979423232, 20506322412604129280, 776179999255323115520, 107455579038104865996800, 4651534843901106606571520, 731092060557632280262082560

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

			The a(1) = 1 through a(3) = 6 chains:
  1  2    6
     2/1  6/1
          6/2
          6/3
          6/2/1
          6/3/1
The a(4) = 40 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

A325617 is the maximal case.
A337070 is the version for superprimorials.
A337074 counts the case with distinct prime multiplicities.
A337105 is the case ending with one.
A000005 counts divisors.
A000142 lists factorial numbers.
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.
Cf. A001013, A001055, A002033, A022559, A124010, A167865, A336941, A336942.

chnsc[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[chnsc[n!]],{n,0,5}]

a(n) = 2*A337105(n) for n > 1.

a(n) = A067824(n!).

a(19)-a(20) from Alois P. Heinz, Aug 23 2020

A342085 Number of decreasing chains of distinct superior divisors starting with n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 4, 2, 2, 1, 11, 2, 2, 3, 4, 1, 7, 1, 10, 2, 2, 2, 15, 1, 2, 2, 10, 1, 6, 1, 4, 5, 2, 1, 26, 2, 5, 2, 4, 1, 11, 2, 10, 2, 2, 1, 21, 1, 2, 5, 20, 2, 6, 1, 4, 2, 7, 1, 39, 1, 2, 5, 4, 2, 6, 1, 23, 6, 2, 1

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
These chains have first-quotients (in analogy with first-differences) that are term-wise less than or equal to their decapitation (maximum element removed). Equivalently, x <= y^2 for all adjacent x, y. For example, the divisor chain q = 24/8/4/2 has first-quotients (3,2,2), which are less than or equal to (8,4,2), so q is counted under a(24).
Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.

			The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12      16        20       24         30       32
     4/2  8/4    12/4    16/4      20/5     24/6       30/6     32/8
          8/4/2  12/6    16/8      20/10    24/8       30/10    32/16
                 12/4/2  16/4/2    20/10/5  24/12      30/15    32/8/4
                 12/6/3  16/8/4             24/6/3     30/6/3   32/16/4
                         16/8/4/2           24/8/4     30/10/5  32/16/8
                                            24/12/4    30/15/5  32/8/4/2
                                            24/12/6             32/16/4/2
                                            24/8/4/2            32/16/8/4
                                            24/12/4/2           32/16/8/4/2
                                            24/12/6/3
The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
  2  4    8      12     16       20     24       30     32
     2*2  4*2    4*3    4*4      5*4    6*4      6*5    8*4
          2*2*2  6*2    8*2      10*2   8*3      10*3   16*2
                 2*2*3  2*2*4    5*2*2  12*2     15*2   4*2*4
                 3*2*2  4*2*2           3*2*4    3*2*5  4*4*2
                        2*2*2*2         4*2*3    5*2*3  8*2*2
                                        4*3*2    5*3*2  2*2*2*4
                                        6*2*2           2*2*4*2
                                        2*2*2*3         4*2*2*2
                                        2*2*3*2         2*2*2*2*2
                                        3*2*2*2

Alois P. Heinz, Table of n, a(n) for n = 1..65536

The restriction to powers of 2 is A045690.
The inferior version is A337135.
The strictly inferior version is A342083.
The strictly superior version is A342084.
The additive version is A342094, with strict case A342095.
The additive version not allowing equality is A342098.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341676.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341673.
Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.

a:= proc(n) option remember; 1+add(`if`(d>=n/d,
      a(d), 0), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..128);  # Alois P. Heinz, Jun 24 2021

cmo[n_]:=Prepend[Prepend[#,n]&/@Join@@cmo/@Select[Most[Divisors[n]],#>=n/#&],{n}];
Table[Length[cmo[n]],{n,100}]

a(2^n) = A045690(n).

A337255 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 5, 7, 3, 1, 1, 1, 3, 2, 1, 3, 2, 1, 4, 6, 4, 1, 1, 1, 1, 5, 7, 3, 1, 1, 1, 5, 7, 3, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 15, 13, 4, 1, 2, 1, 1, 3, 2, 1, 3, 3, 1, 1, 5, 7, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Aug 23 2020

			Sequence of rows begins:
     1: {1}           16: {1,4,6,4,1}
     2: {1,1}         17: {1,1}
     3: {1,1}         18: {1,5,7,3}
     4: {1,2,1}       19: {1,1}
     5: {1,1}         20: {1,5,7,3}
     6: {1,3,2}       21: {1,3,2}
     7: {1,1}         22: {1,3,2}
     8: {1,3,3,1}     23: {1,1}
     9: {1,2,1}       24: {1,7,15,13,4}
    10: {1,3,2}       25: {1,2,1}
    11: {1,1}         26: {1,3,2}
    12: {1,5,7,3}     27: {1,3,3,1}
    13: {1,1}         28: {1,5,7,3}
    14: {1,3,2}       29: {1,1}
    15: {1,3,2}       30: {1,7,12,6}
Row n = 24 counts the following chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/6/2/1   24/12/4/2/1
      24/3   24/4/1   24/6/3/1   24/12/6/2/1
      24/4   24/4/2   24/8/2/1   24/12/6/3/1
      24/6   24/6/1   24/8/4/1
      24/8   24/6/2   24/8/4/2
      24/12  24/6/3   24/12/2/1
             24/8/1   24/12/3/1
             24/8/2   24/12/4/1
             24/8/4   24/12/4/2
             24/12/1  24/12/6/1
             24/12/2  24/12/6/2
             24/12/3  24/12/6/3
             24/12/4
             24/12/6

Alois P. Heinz, Rows n = 1..5000, flattened

A008480 gives rows ends.
A067824 gives row sums.
A073093 gives row lengths.
A334996 appears to be the case of chains ending with 1.
A337071 is the sum of row n!.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A122651 counts chains of divisors summing to n.
A167865 counts chains of divisors > 1 summing to n.
A251683 counts chains of divisors from n to 1 by length.
A253249 counts nonempty chains of divisors.
A337070 counts chains of divisors starting with A006939(n).
A337256 counts chains of divisors.
Cf. A001221, A002033, A124010, A124433, A337074, A337105, A337107.

b:= proc(n) option remember; expand(x*(1 +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=1..50);  # Alois P. Heinz, Aug 23 2020

chss[n_]:=Prepend[Join@@Table[Prepend[#,n]&/@chss[d],{d,Most[Divisors[n]]}],{n}];
Table[Length[Select[chss[n],Length[#]==k&]],{n,30},{k,1+PrimeOmega[n]}]

A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 9, 4, 0, 1, 14, 45, 52, 20, 0, 1, 28, 183, 496, 655, 420, 105, 0, 1, 58, 633, 2716, 5755, 6450, 3675, 840, 0, 1, 94, 1659, 11996, 46235, 106806, 155869, 145384, 84276, 27720, 3960

Offset: 1

Gus Wiseman, Aug 23 2020

Row n > 1 appears to be row n! of A334996.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    9    4
    0    1   14   45   52   20
    0    1   28  183  496  655  420  105
    0    1   58  633 2716 5755 6450 3675  840
Row n = 4 counts the following chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1

A097805 is the restriction to powers of 2.
A325617 is the maximal case.
A337105 gives row sums.
A337106 is column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
A074206 counts chains of divisors from n to 1.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A076716 counts factorizations of factorial numbers.
A253249 counts chains of divisors.
A337071 counts chains starting with n!.
Cf. A022559, A124010, A167865, A251683, A337070, A337074.

b:= proc(n) option remember; expand(x*(`if`(n=1, 1, 0) +
      add(b(d), d=numtheory[divisors](n) minus {n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n!)):
seq(T(n), n=1..10);  # Alois P. Heinz, Aug 23 2020

nv=5;
chnsc[n_]:=Select[Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],n]}],{n}],MemberQ[#,1]&];
Table[Length[Select[chnsc[n!],Length[#]==k&]],{n,nv},{k,1+PrimeOmega[n!]}]

A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 4, 5, 2, 1, 2, 1, 1, 4, 6, 4, 1, 1, 3, 3, 1, 1, 4, 5, 2, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 4, 5, 2, 1, 4, 5, 2, 1, 5, 10, 10, 5, 1, 1, 2, 1, 1, 6, 12, 10, 3, 1, 2, 1, 1, 6, 12, 10, 3, 1, 4, 5, 2, 1, 4, 5, 2

Offset: 1

Gus Wiseman, May 01 2021

			Triangle begins:
   1:  1  1
   2:  1  2  1
   3:  1  2  1
   4:  1  3  3  1
   5:  1  2  1
   6:  1  4  5  2
   7:  1  2  1
   8:  1  4  6  4  1
   9:  1  3  3  1
  10:  1  4  5  2
  11:  1  2  1
  12:  1  6 12 10  3
  13:  1  2  1
  14:  1  4  5  2
  15:  1  4  5  2
  16:  1  5 10 10  5  1
For example, row n = 12 counts the following chains:
  ()  (1)   (2/1)   (4/2/1)   (12/4/2/1)
      (2)   (3/1)   (6/2/1)   (12/6/2/1)
      (3)   (4/1)   (6/3/1)   (12/6/3/1)
      (4)   (4/2)   (12/2/1)
      (6)   (6/1)   (12/3/1)
      (12)  (6/2)   (12/4/1)
            (6/3)   (12/4/2)
            (12/1)  (12/6/1)
            (12/2)  (12/6/2)
            (12/3)  (12/6/3)
            (12/4)
            (12/6)

Column k = 1 is A000005.
Row ends are A008480.
Row lengths are A073093.
Column k = 2 is A238952.
The case from n to 1 is A334996 or A251683 (row sums: A074206).
A non-strict version is A334997 (transpose: A077592).
The case starting with n is A337255 (row sums: A067824).
Row sums are A337256 (nonempty: A253249).
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A097805 counts compositions by sum and length.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A163767 counts length n - 1 chains of divisors of n.
A167865 counts strict chains of divisors > 1 summing to n.
A337070 counts strict chains of divisors starting with superprimorials.
Cf. A002033, A007425, A007426, A051026, A062319, A143773, A186972, A327527, A337074, A337105, A337107, A343658.

Table[Length[Select[Reverse/@Subsets[Divisors[n],{k}],And@@Divisible@@@Partition[#,2,1]&]],{n,15},{k,0,PrimeOmega[n]+1}]

A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

Original entry on oeis.org

1, 0, -1, 0, -1, 0, -1, 1, 0, -1, 0, -1, 2, 0, -1, 0, -1, 2, -1, 0, -1, 1, 0, -1, 2, 0, -1, 0, -1, 4, -3, 0, -1, 0, -1, 2, 0, -1, 2, 0, -1, 3, -3, 1, 0, -1, 0, -1, 4, -3, 0, -1, 0, -1, 4, -3, 0, -1, 2, 0, -1, 2, 0, -1, 0, -1, 6, -9, 4, 0, -1, 1, 0, -1, 2, 0, -1, 2, -1, 0, -1, 4, -3, 0, -1, 0, -1, 6, -6, 0, -1, 0, -1, 4, -6, 4, -1, 0, -1, 2, 0, -1, 2, 0, -1

Offset: 1

Leroy Quet, Dec 15 2006

Row n has A001222(n)+1 terms. The polynomial P_n(y) = (sum{m=1 to A001222(n)+1} a(n,m)*y^m) is a generalization of the Mobius (Moebius) function, where P_n(1) = A008683(n).
From Gus Wiseman, Aug 24 2020: (Start)
Up to sign, also the number of strict length-k chains of divisors from n to 1, 1 <= k <= 1 + A001222(n). For example, row n = 36 counts the following chains (empty column indicated by dot):
  .  36/1  36/2/1   36/4/2/1   36/12/4/2/1
           36/3/1   36/6/2/1   36/12/6/2/1
           36/4/1   36/6/3/1   36/12/6/3/1
           36/6/1   36/9/3/1   36/18/6/2/1
           36/9/1   36/12/2/1  36/18/6/3/1
           36/12/1  36/12/3/1  36/18/9/3/1
           36/18/1  36/12/4/1
                    36/12/6/1
                    36/18/2/1
                    36/18/3/1
                    36/18/6/1
                    36/18/9/1
(End)

			1/(zeta(x) - 1 + 1/y) = y - y^2/2^x - y^2/3^x + ( - y^2 + y^3)/4^x - y^2/5^x + ( - y^2 + 2y^3)/6^x - y^2/7^x + ...
From _Gus Wiseman_, Aug 24 2020: (Start)
The sequence of rows begins:
     1: 1              16: 0 -1 3 -3 1     31: 0 -1
     2: 0 -1           17: 0 -1            32: 0 -1 4 -6 4 -1
     3: 0 -1           18: 0 -1 4 -3       33: 0 -1 2
     4: 0 -1 1         19: 0 -1            34: 0 -1 2
     5: 0 -1           20: 0 -1 4 -3       35: 0 -1 2
     6: 0 -1 2         21: 0 -1 2          36: 0 -1 7 -12 6
     7: 0 -1           22: 0 -1 2          37: 0 -1
     8: 0 -1 2 -1      23: 0 -1            38: 0 -1 2
     9: 0 -1 1         24: 0 -1 6 -9 4     39: 0 -1 2
    10: 0 -1 2         25: 0 -1 1          40: 0 -1 6 -9 4
    11: 0 -1           26: 0 -1 2          41: 0 -1
    12: 0 -1 4 -3      27: 0 -1 2 -1       42: 0 -1 6 -6
    13: 0 -1           28: 0 -1 4 -3       43: 0 -1
    14: 0 -1 2         29: 0 -1            44: 0 -1 4 -3
    15: 0 -1 2         30: 0 -1 6 -6       45: 0 -1 4 -3
(End)

Mohammad K. Azarian, A Double Sum, Problem 440, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. Solution published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.

A008480 gives rows ends (up to sign).
A008683 gives row sums (the Moebius function).
A073093 gives row lengths.
A074206 gives unsigned row sums.
A097805 is the restriction to powers of 2 (up to sign).
A251683 is the unsigned version with zeros removed.
A334996 is the unsigned version (except with a(1) = 0).
A334997 is an unsigned non-strict version.
A337107 is the restriction to factorial numbers.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A337105 counts strict chains of divisors from n! to 1.
Cf. A000005, A001221, A002033, A124010, A167865.

f[l_List] := Block[{n = Length[l] + 1, c},c = Plus @@ Last /@ FactorInteger[n];Append[l, Prepend[ -Plus @@ Pick[PadRight[ #, c] & /@ l, Mod[n, Range[n - 1]], 0],0]]];Nest[f, {{1}}, 34] // Flatten(* Ray Chandler, Feb 13 2007 *)
chnsc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]];
Table[(-1)^k*Length[Select[chnsc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* Gus Wiseman, Aug 24 2020 *)

a(1,1)=1. a(n,1) = 0 for n>=2. a(n,m+1) = -sum{k|n,k < n} a(k,m), where, for the purpose of this sum, a(k,m) = 0 if m > A001222(k)+1.

Extended by Ray Chandler, Feb 13 2007

A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 0, 14, 0, 384, 0, 0, 0, 21077680, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

The support appears to be {0, 1, 2, 4, 6, 10}.

			The a(4) = 14 chains:
  24/1
  24/2/1
  24/3/1
  24/4/1
  24/8/1
  24/12/1
  24/4/2/1
  24/8/2/1
  24/8/4/1
  24/12/2/1
  24/12/3/1
  24/12/4/1
  24/8/4/2/1
  24/12/4/2/1

A336867 appears to be the positions of zeros.
A336868 is the characteristic function (image under A057427).
A336942 is the version for superprimorials (n > 1).
A337105 does not require distinct prime multiplicities.
A337074 does not require chains to end with 1.
A337075 is the version for chains not containing n!.
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.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336425 counts divisible pairs of divisors of n!, both in A130091.
A336571 counts chains of divisors 1 < d < n using A130091.
A337071 counts chains of divisors starting with n!.
Cf. A002033, A022559, A071626, A098859, A124010, A167865, A327523, A336424, A336941.

strchns[n_]:=If[n==1,1,If[!UnsameQ@@Last/@FactorInteger[n],0,Sum[strchns[d],{d,Select[DeleteCases[Divisors[n],n],UnsameQ@@Last/@FactorInteger[#]&]}]]];
Table[strchns[n!],{n,0,8}]

a(n) = A337075(n) whenever A337075(n) != 0.
a(n) = A337074(n)/2 for n > 1.
a(n) = A336423(n!).

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

A337071 Number of strict chains of divisors starting with n!.

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A342085 Number of decreasing chains of distinct superior divisors starting with n.

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A337255 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors starting with n.

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A337107 Irregular triangle read by rows where T(n,k) is the number of strict length-k chains of divisors from n! to 1.

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A343662 Irregular triangle read by rows where T(n,k) is the number of strict length k chains of divisors of n, 0 <= k <= Omega(n) + 1.

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A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

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A337104 Number of strict chains of divisors from n! to 1 using terms of A130091 (numbers with distinct prime multiplicities).

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A337106 Number of nontrivial divisors of n!.

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