A334997 -id:A334997 - cp's OEIS frontend

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

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

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).

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}]

A342530 Number of strict chains of divisors ending with n and having distinct first quotients.

Original entry on oeis.org

1, 2, 2, 3, 2, 6, 2, 6, 3, 6, 2, 12, 2, 6, 6, 9, 2, 12, 2, 12, 6, 6, 2, 28, 3, 6, 6, 12, 2, 26, 2, 14, 6, 6, 6, 31, 2, 6, 6, 28, 2, 26, 2, 12, 12, 6, 2, 52, 3, 12, 6, 12, 2, 28, 6, 28, 6, 6, 2, 66, 2, 6, 12, 25, 6, 26, 2, 12, 6, 26, 2, 76, 2, 6, 12, 12, 6, 26

Offset: 1

Gus Wiseman, Mar 25 2021

nonn

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the quotients of (6,3,1) are (1/2,1/3).

			The a(1) = 1 through a(12) = 12 chains (reversed):
  1  2    3    4    5    6      7    8      9    10      11    12
     2/1  3/1  4/1  5/1  6/1    7/1  8/1    9/1  10/1    11/1  12/1
               4/2       6/2         8/2    9/3  10/2          12/2
                         6/3         8/4         10/5          12/3
                         6/2/1       8/2/1       10/2/1        12/4
                         6/3/1       8/4/1       10/5/1        12/6
                                                               12/2/1
                                                               12/3/1
                                                               12/4/1
                                                               12/4/2
                                                               12/6/1
                                                               12/6/2
Not counted under a(12) are: 12/4/2/1, 12/6/2/1, 12/6/3, 12/6/3/1.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version for weakly increasing first quotients is A057567.
The version for equal first quotients is A169594.
The case of chains starting with 1 is A254578.
The version for strictly increasing first quotients is A342086.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A067824 counts strict chains of divisors ending with n.
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.
A342495/A342529 count compositions with equal/distinct quotients.
A342496/A342514 count partitions with equal/distinct quotients.
A342515/A342520 count strict partitions with equal/distinct quotients.
A342522/A342521 rank partitions with equal/distinct quotients.
Cf. A000009, A003238, A003242, A122651, A179254, A318991, A318992, A325545.

cmi[n_]:=Prepend[Prepend[#,n]&/@Join@@cmi/@Most[Divisors[n]],{n}];
Table[Length[Select[cmi[n],UnsameQ@@Divide@@@Partition[#,2,1]&]],{n,100}]

a(n) = Sum_{d|n} A254578(d). - Ridouane Oudra, Jun 17 2025

A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 84 are {1,1,2,4}, with first quotients (1,2,2), so 84 is not in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   27: {2,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   48: {1,1,1,1,2}
   50: {1,3,3}
   54: {1,2,2,2}
   56: {1,1,1,4}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

For differences instead of quotients we have A325456 (count: A240027).
For multiplicities (prime signature) instead of quotients we have A334965.
The version counting strict divisor chains is A342086.
These partitions are counted by A342498 (strict: A342517, ordered: A342493).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A048767, A056239, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A334997, A342530.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 150 are {1,2,3,3}, with first quotients (2,3/2,1), so 150 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   20: {1,1,3}
   24: {1,1,1,2}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

For multiplicities (prime signature) instead of quotients we have A304686.
For differences instead of quotients we have A325457 (count: A320470).
The version counting strict divisor chains is A342086.
These partitions are counted by A342499 (strict: A342518, ordered: A342494).
The strictly increasing version is A342524.
The weakly decreasing version is A342526.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A056239, A067824, A112798, A124010, A130091, A169594, A253249, A325351, A325352, A325405, A334997, A342530.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A343935 Number of ways to choose a multiset of n divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 84, 8, 165, 55, 286, 12, 6188, 14, 680, 816, 4845, 18, 33649, 20, 53130, 2024, 2300, 24, 2629575, 351, 3654, 4060, 237336, 30, 10295472, 32, 435897, 7140, 7770, 8436, 177232627, 38, 10660, 11480, 62891499, 42, 85900584, 44, 1906884, 2118760

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 multisets:
  {1}  {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}
       {1,2}  {1,1,3}  {1,1,1,2}  {1,1,1,1,5}
       {2,2}  {1,3,3}  {1,1,1,4}  {1,1,1,5,5}
              {3,3,3}  {1,1,2,2}  {1,1,5,5,5}
                       {1,1,2,4}  {1,5,5,5,5}
                       {1,1,4,4}  {5,5,5,5,5}
                       {1,2,2,2}
                       {1,2,2,4}
                       {1,2,4,4}
                       {1,4,4,4}
                       {2,2,2,2}
                       {2,2,2,4}
                       {2,2,4,4}
                       {2,4,4,4}
                       {4,4,4,4}

Diagonal n = k of A343658.
Choosing n divisors of n - 1 gives A343936.
The version for chains of divisors is A343939.
A000005 counts divisors.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998 = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
A146291 counts divisors of n with k prime factors (with multiplicity).
A253249 counts nonempty chains of divisors of n.
Strict chains of divisors:
- A067824 counts strict chains of divisors starting with n.
- A074206 counts strict chains of divisors from n to 1.
- A251683 counts strict length k + 1 chains of divisors from n to 1.
- A334996 counts strict length-k chains of divisors from n to 1.
- A337255 counts strict length-k chains of divisors starting with n.
- A337256 counts strict chains of divisors of n.
- A343662 counts strict length-k chains of divisors.
Cf. A062319, A143773, A163767, A176029, A327527, A334997, A343661.

multchoo[n_,k_]:=Binomial[n+k-1,k];
Table[multchoo[DivisorSigma[0,n],n],{n,25}]

from math import comb
from sympy import divisor_count
def A343935(n): return comb(divisor_count(n)+n-1,n) # Chai Wah Wu, Jul 05 2024

a(n) = ((sigma(n), n)) = binomial(sigma(n) + n - 1, n) where sigma = A000005 and binomial = A007318.

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

A342526 Heinz numbers of integer partitions with weakly decreasing first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 81, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Mar 23 2021

nonn

Also called log-concave-down partitions.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

			The prime indices of 294 are {1,2,4,4}, with first quotients (2,2,1), so 294 is in the sequence.
Most small numbers are in the sequence, but the sequence of non-terms together with their prime indices begins:
   12: {1,1,2}
   20: {1,1,3}
   24: {1,1,1,2}
   28: {1,1,4}
   36: {1,1,2,2}
   40: {1,1,1,3}
   44: {1,1,5}
   45: {2,2,3}
   48: {1,1,1,1,2}
   52: {1,1,6}
   56: {1,1,1,4}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   68: {1,1,7}
   72: {1,1,1,2,2}
   76: {1,1,8}
   78: {1,2,6}
   80: {1,1,1,1,3}
   84: {1,1,2,4}

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.
Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

The version counting strict divisor chains is A057567.
For multiplicities (prime signature) instead of quotients we have A242031.
For differences instead of quotients we have A325361 (count: A320466).
These partitions are counted by A342513 (strict: A342519, ordered: A069916).
The weakly increasing version is A342523.
The strictly decreasing version is A342525.
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A002843 counts compositions with all adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A167865 counts strict chains of divisors > 1 summing to n.
A318991/A318992 rank reversed partitions with/without integer quotients.
Cf. A048767, A056239, A067824, A112798, A238710, A253249, A325351, A325352, A325405, A334997, A342086, A342191.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],GreaterEqual@@Divide@@@Reverse/@Partition[primeptn[#],2,1]&]

A343939 Number of n-chains of divisors of n.

Original entry on oeis.org

1, 3, 4, 15, 6, 49, 8, 165, 55, 121, 12, 1183, 14, 225, 256, 4845, 18, 3610, 20, 4851, 484, 529, 24, 73125, 351, 729, 4060, 12615, 30, 29791, 32, 435897, 1156, 1225, 1296, 494209, 38, 1521, 1600, 505981, 42, 79507, 44, 46575, 49726, 2209, 48

Offset: 1

Gus Wiseman, May 05 2021

nonn

			The a(1) = 1 through a(5) = 6 chains:
  (1)  (1/1)  (1/1/1)  (1/1/1/1)  (1/1/1/1/1)
       (2/1)  (3/1/1)  (2/1/1/1)  (5/1/1/1/1)
       (2/2)  (3/3/1)  (2/2/1/1)  (5/5/1/1/1)
              (3/3/3)  (2/2/2/1)  (5/5/5/1/1)
                       (2/2/2/2)  (5/5/5/5/1)
                       (4/1/1/1)  (5/5/5/5/5)
                       (4/2/1/1)
                       (4/2/2/1)
                       (4/2/2/2)
                       (4/4/1/1)
                       (4/4/2/1)
                       (4/4/2/2)
                       (4/4/4/1)
                       (4/4/4/2)
                       (4/4/4/4)

Diagonal n = k - 1 of the array A077592.
Chains of length n - 1 are counted by A163767.
Diagonal n = k of the array A334997.
The version counting all multisets of divisors (not just chains) is A343935.
A000005(n) counts divisors of n.
A067824(n) counts strict chains of divisors starting with n.
A074206(n) counts strict chains of divisors from n to 1.
A146291(n,k) counts divisors of n with k prime factors (with multiplicity).
A251683(n,k-1) counts strict k-chains of divisors from n to 1.
A253249(n) counts nonempty chains of divisors of n.
A334996(n,k) counts strict k-chains of divisors from n to 1.
A337255(n,k) counts strict k-chains of divisors starting with n.
A343658(n,k) counts k-multisets of divisors of n.
A343662(n,k) counts strict k-chains of divisors of n (row sums: A337256).
Cf. A062319, A066959, A143773, A176029, A327527, A343656.

Table[Length[Select[Tuples[Divisors[n],n],OrderedQ[#]&&And@@Divisible@@@Reverse/@Partition[#,2,1]&]],{n,10}]

cp's OEIS Frontend

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

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

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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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A342530 Number of strict chains of divisors ending with n and having distinct first quotients.

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A342524 Heinz numbers of integer partitions with strictly increasing first quotients.

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A342525 Heinz numbers of integer partitions with strictly decreasing first quotients.

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A343935 Number of ways to choose a multiset of n divisors of n.

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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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