A067824 -id:A067824 - cp's OEIS frontend

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 2, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 8, 2, 4, 4, 8, 2, 10, 2, 6, 6, 4, 2, 12, 2, 6, 4, 6, 2, 10, 4, 8, 4, 4, 2, 14, 2, 4, 6, 6, 4, 10, 2, 6, 4, 8, 2, 16, 2, 4, 6, 6, 4, 10, 2, 12, 4, 4, 2, 14

Offset: 1

Gus Wiseman, Mar 05 2021

nonn

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.

			The chains for n = 1, 2, 6, 12, 24, 42, 48:
   1    2      6        12        24        42          48
        2/1    6/1      12/1      24/1      42/1        48/1
               6/2      12/2      24/2      42/2        48/2
               6/2/1    12/3      24/3      42/3        48/3
                        12/2/1    24/4      42/6        48/4
                        12/3/1    24/2/1    42/2/1      48/6
                                  24/3/1    42/3/1      48/2/1
                                  24/4/1    42/6/1      48/3/1
                                            42/6/2      48/4/1
                                            42/6/2/1    48/6/1
                                                        48/6/2
                                                        48/6/2/1

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A018819.
Not requiring strict inferiority gives A067824.
The weakly inferior version is twice A337135.
The case ending with 1 is counted by A342083.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors, with sum A000203.
A001055 counts factorizations.
A003238 counts chains of divisors 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 ordered factorizations.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334997 counts chains of divisors of n by length.
Cf. A002033, A006530, A018819, A020639, A337105, A341674, A342086, A342094, A342095, A342097.

cem[n_]:=Prepend[Prepend[#,n]&/@Join@@cem/@Most[Divisors[n]],{n}];
Table[Length[Select[cem[n],And@@Thread[Divide@@@Partition[#,2,1]>Rest[#]]&]],{n,30}]

For n > 1, a(n) = 2*A342083(n).

A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

Original entry on oeis.org

1, 2, 3, 10, 5, 36, 7, 120, 45, 100, 11, 936, 13, 196, 225, 3876, 17, 3078, 19, 4200, 441, 484, 23, 62400, 325, 676, 3654, 11368, 29, 27000, 31, 376992, 1089, 1156, 1225, 443556, 37, 1444, 1521, 459200, 41, 74088, 43, 43560, 46575, 2116, 47, 11995200, 1225

Offset: 1

Paul D. Hanna, Aug 04 2009

nonn

Also the number of length n - 1 chains of divisors of n. - Gus Wiseman, May 07 2021

			Successive Dirichlet self-convolutions of the all 1's sequence begin:
(1),1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,... (A000012)
1,(2),2,3,2,4,2,4,3,4,2,6,2,4,4,5,... (A000005)
1,3,(3),6,3,9,3,10,6,9,3,18,3,9,9,15,... (A007425)
1,4,4,(10),4,16,4,20,10,16,4,40,4,16,16,35,... (A007426)
1,5,5,15,(5),25,5,35,15,25,5,75,5,25,25,70,... (A061200)
1,6,6,21,6,(36),6,56,21,36,6,126,6,36,36,126,... (A034695)
1,7,7,28,7,49,(7),84,28,49,7,196,7,49,49,210,... (A111217)
1,8,8,36,8,64,8,(120),36,64,8,288,8,64,64,330,... (A111218)
1,9,9,45,9,81,9,165,(45),81,9,405,9,81,81,495,... (A111219)
1,10,10,55,10,100,10,220,55,(100),10,550,10,100,... (A111220)
1,11,11,66,11,121,11,286,66,121,(11),726,11,121,... (A111221)
1,12,12,78,12,144,12,364,78,144,12,(936),12,144,... (A111306)
...
where the main diagonal forms this sequence.
From _Gus Wiseman_, May 07 2021: (Start)
The a(1) = 1 through a(5) = 5 chains of divisors:
  ()  (1)  (1/1)  (1/1/1)  (1/1/1/1)
      (2)  (3/1)  (2/1/1)  (5/1/1/1)
           (3/3)  (2/2/1)  (5/5/1/1)
                  (2/2/2)  (5/5/5/1)
                  (4/1/1)  (5/5/5/5)
                  (4/2/1)
                  (4/2/2)
                  (4/4/1)
                  (4/4/2)
                  (4/4/4)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Enrique Pérez Herrero)

Main diagonal of A077592.
Diagonal n = k + 1 of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005 counts divisors.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A146291 counts divisors of n with k prime factors (with multiplicity).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts nonempty strict chains of divisors of n.
A251683/A334996 count strict nonempty length-k divisor chains from n to 1.
A337255 counts strict length-k chains of divisors starting with n.
A339564 counts factorizations with a selected factor.
A343662 counts strict length-k chains of divisors (row sums: A337256).
Cf. A002033, A007425, A008480, A018818, A062319, A066959, A186972, A327527, A337105, A337107, A343658.
Cf. A060690.

Table[Times@@(Binomial[#+n-1,n-1]&/@FactorInteger[n][[All,2]]),{n,1,50}] (* Enrique Pérez Herrero, Dec 25 2013 *)

{a(n,m=n)=if(n==1,1,if(m==1,1,sumdiv(n,d,a(d,1)*a(n/d,m-1))))}

from math import prod, comb
from sympy import factorint
def A163767(n): return prod(comb(n+e-1,e) for e in factorint(n).values()) # Chai Wah Wu, Jul 05 2024

a(p) = p for prime p.
a(n) = n^k when n is the product of k distinct primes (conjecture).
a(n) = n-th term of the n-th Dirichlet self-convolution of the all 1's sequence.
a(2^n) = A060690(n). - Alois P. Heinz, Jun 12 2024

A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 2, 1, 1, 6, 5, 10, 3, 4, 1, 1, 7, 6, 15, 4, 10, 2, 1, 1, 8, 7, 21, 5, 20, 3, 4, 1, 1, 9, 8, 28, 6, 35, 4, 10, 3, 1, 1, 10, 9, 36, 7, 56, 5, 20, 6, 4, 1, 1, 11, 10, 45, 8, 84, 6, 35, 10, 10, 2, 1

Offset: 1

Gus Wiseman, Apr 29 2021

First differs from A343656 at A(4,2) = 6, A343656(4,2) = 5.

As a triangle, T(n,k) = number of ways to choose a multiset of n - k divisors of k.

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
  n=1:  1   1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8   9
  n=3:  1   2   3   4   5   6   7   8   9
  n=4:  1   3   6  10  15  21  28  36  45
  n=5:  1   2   3   4   5   6   7   8   9
  n=6:  1   4  10  20  35  56  84 120 165
  n=7:  1   2   3   4   5   6   7   8   9
  n=8:  1   4  10  20  35  56  84 120 165
  n=9:  1   3   6  10  15  21  28  36  45
Triangle begins:
   1
   1   1
   1   2   1
   1   3   2   1
   1   4   3   3   1
   1   5   4   6   2   1
   1   6   5  10   3   4   1
   1   7   6  15   4  10   2   1
   1   8   7  21   5  20   3   4   1
   1   9   8  28   6  35   4  10   3   1
   1  10   9  36   7  56   5  20   6   4   1
   1  11  10  45   8  84   6  35  10  10   2   1
For example, row n = 6 counts the following multisets:
  {1,1,1,1,1}  {1,1,1,1}  {1,1,1}  {1,1}  {1}  {}
               {1,1,1,2}  {1,1,3}  {1,2}  {5}
               {1,1,2,2}  {1,3,3}  {1,4}
               {1,2,2,2}  {3,3,3}  {2,2}
               {2,2,2,2}           {2,4}
                                   {4,4}
Note that for n = 6, k = 4 in the triangle, the two multisets {1,4} and {2,2} represent the same divisor 4, so they are only counted once under A343656(4,2) = 5.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)

Row k = 1 of the array is A000005.
Column n = 4 of the array is A000217.
Column n = 6 of the array is A000292.
Row k = 2 of the array is A184389.
The distinct products of these multisets are counted by A343656.
Antidiagonal sums of the array (or row sums of the triangle) are A343661.
A000312 = n^n.
A009998(n,k) = n^k (as an array, offset 1).
A007318 counts k-sets of elements of {1..n}.
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A062319, A067824, A143773, A146291, A176029, A285572, A326077, A327527, A334996, A343652, A343657.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,k],n-k],{n,10},{k,n}]

A(n,k) = binomial(numdiv(n) + k - 1, k)
{ for(n=1, 9, for(k=0, 8, print1(A(n,k), ", ")); print ) } \\ Andrew Howroyd, Jan 11 2024

A(n,k) = ((A000005(n), k)) = A007318(A000005(n) + k - 1, k).

T(n,k) = ((A000005(k), n - k)) = A007318(A000005(k) + n - k - 1, n - k).

A316782 Number of achiral tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 13 2018

nonn

A factorization of n is a finite nonempty multiset of positive integers greater than 1 with product n. An achiral tree-factorization of n is either (case 1) the number n itself or (case 2) a finite constant multiset of two or more achiral tree-factorizations, one of each factor in a factorization of n.
a(n) is also the number of ways to write n as a left-nested power-tower ((a^b)^c)^... of positive integers greater than one. For example, the a(64) = 6 ways are 64, 8^2, 4^3, 2^6, (2^3)^2, (2^2)^3.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(1296) = 4 achiral tree-factorizations are 1296, (36*36), (6*6*6*6), ((6*6)*(6*6)).

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A001055, A001597, A003238, A067824, A089723, A214577, A281118, A289078, A292504, A294336.

a[n_]:=1+Sum[a[d],{d,n^(1/Rest[Divisors[GCD@@FactorInteger[n][[All,2]]]])}];
Array[a,100]

a(n)={my(z, e=ispower(n,,&z)); 1 + if(e, sumdiv(e, d, if(dAndrew Howroyd, Nov 18 2018

a(n) = 1 + Sum_{n = d^k, k>1} a(d).

a(p^n) = A067824(n) for prime p. - Andrew Howroyd, Nov 18 2018

A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

Original entry on oeis.org

1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions.

			The first 4 terms count the following multifunctions.
o,
o[o],
o[o,o], o[o[o]], o[o][o],
o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)

A337256 Number of strict chains of divisors of n.

Original entry on oeis.org

2, 4, 4, 8, 4, 12, 4, 16, 8, 12, 4, 32, 4, 12, 12, 32, 4, 32, 4, 32, 12, 12, 4, 80, 8, 12, 16, 32, 4, 52, 4, 64, 12, 12, 12, 104, 4, 12, 12, 80, 4, 52, 4, 32, 32, 12, 4, 192, 8, 32, 12, 32, 4, 80, 12, 80, 12, 12, 4, 176, 4, 12, 32, 128, 12, 52, 4, 32, 12, 52

Offset: 1

Gus Wiseman, Aug 23 2020

nonn

			The a(n) chains for n = 1, 2, 4, 6, 8 (empty chains shown as 0):
  0  0    0      0      0
  1  1    1      1      1
     2    2      2      2
     2/1  4      3      4
          2/1    6      8
          4/1    2/1    2/1
          4/2    3/1    4/1
          4/2/1  6/1    4/2
                 6/2    8/1
                 6/3    8/2
                 6/2/1  8/4
                 6/3/1  4/2/1
                        8/2/1
                        8/4/1
                        8/4/2
                        8/4/2/1

A067824 is the case of chains starting with n (or ending with 1).
A074206 is the case of chains from n to 1.
A253249 is the nonempty case.
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
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.
A334996 appears to count chains of divisors from n to 1 by length.
A337070 counts chains of divisors starting with A006939(n).
A337071 counts chains of divisors starting with n!.
A337255 counts chains of divisors starting with n by length.
Cf. A001221, A002033, A008480, A124010, A251683, A337105, A337107.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Divisors[n],!(Divisible[#1,#2]||Divisible[#2,#1])&]],{n,10}]

a(n) = A253249(n) + 1.

A343652 Number of maximal pairwise coprime sets of divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2

Offset: 1

Gus Wiseman, Apr 25 2021

nonn

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:
  {6}    {30}     {6}    {30}     {30}
  {12}   {2,15}   {12}   {60}     {60}
  {2,3}  {3,10}   {18}   {2,15}   {120}
  {3,4}  {5,6}    {36}   {3,10}   {2,15}
         {2,3,5}  {2,3}  {3,20}   {3,10}
                  {2,9}  {4,15}   {3,20}
                  {3,4}  {5,6}    {3,40}
                  {4,9}  {5,12}   {4,15}
                         {2,3,5}  {5,6}
                         {3,4,5}  {5,12}
                                  {5,24}
                                  {8,15}
                                  {2,3,5}
                                  {3,4,5}
                                  {3,5,8}

			The a(n) sets for n = 12, 30, 36, 60, 120:
  {1,6}    {1,30}     {1,6}    {1,30}     {1,30}
  {1,12}   {1,2,15}   {1,12}   {1,60}     {1,60}
  {1,2,3}  {1,3,10}   {1,18}   {1,2,15}   {1,120}
  {1,3,4}  {1,5,6}    {1,36}   {1,3,10}   {1,2,15}
           {1,2,3,5}  {1,2,3}  {1,3,20}   {1,3,10}
                      {1,2,9}  {1,4,15}   {1,3,20}
                      {1,3,4}  {1,5,6}    {1,3,40}
                      {1,4,9}  {1,5,12}   {1,4,15}
                               {1,2,3,5}  {1,5,6}
                               {1,3,4,5}  {1,5,12}
                                          {1,5,24}
                                          {1,8,15}
                                          {1,2,3,5}
                                          {1,3,4,5}
                                          {1,3,5,8}

The case of pairs is A063647.
The case of triples is A066620.
The non-maximal version counting empty sets and singletons is A225520.
The non-maximal version with no 1's is A343653.
The non-maximal version is A343655.
The version for subsets of {1..n} is A343659.
The case without 1's or singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A084422, A187106, A276187, and A320426 count pairwise coprime sets.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A324837 counts minimal subsets of {1...n} with least common multiple n.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

a(n) = A343660(n) + A005361(n).

A337070 Number of strict chains of divisors starting with the superprimorial A006939(n).

Original entry on oeis.org

1, 2, 16, 1208, 1383936, 32718467072, 20166949856488576, 391322675415566237681536

Offset: 0

Gus Wiseman, Aug 15 2020

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

			The a(0) = 1 through a(2) = 16 chains:
  1  2    12
     2/1  12/1
          12/2
          12/3
          12/4
          12/6
          12/2/1
          12/3/1
          12/4/1
          12/4/2
          12/6/1
          12/6/2
          12/6/3
          12/4/2/1
          12/6/2/1
          12/6/3/1

A022915 is the maximal case.
A076954 can be used instead of A006939 (cf. A307895, A325337).
A336571 is the case with distinct prime multiplicities.
A336941 is the case ending with 1.
A337071 is the version for factorials.
A000005 counts divisors.
A000142 counts divisors of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A253249 counts chains of divisors.
A317829 counts factorizations of superprimorials.
Cf. A001055, A002033, A167865, A181818, A336420, A336423, A336942, A337074.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
chnsc[n_]:=If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]];
Table[Length[chnsc[chern[n]]],{n,0,3}]

a(n) = 2*A336941(n) for n > 0.

a(n) = A067824(A006939(n)).

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

Original entry on oeis.org

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).

cp's OEIS Frontend

A342087 Number of chains of divisors starting with n and having no adjacent parts x <= y^2.

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A163767 a(n) = tau_{n}(n) = number of ordered n-factorizations of n.

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A343658 Array read by antidiagonals where A(n,k) is the number of ways to choose a multiset of k divisors of n.

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A316782 Number of achiral tree-factorizations of n.

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A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

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A337256 Number of strict chains of divisors of n.

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A343652 Number of maximal pairwise coprime sets of divisors of n.

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