A067824 -id:A067824 - cp's OEIS frontend

A317853 a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).

1, 1, 1, 1, 2, 2, 5, 6, 11, 14, 23, 26, 51, 70, 114, 147, 237, 314, 516, 715, 1118, 1549, 2353, 3252, 5011, 7235, 10724, 15142, 22504, 32506, 47770, 69173, 100980, 146657, 212504, 308563, 448256, 658037, 946166, 1373739, 1988283, 2919185, 4197886, 6118850

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=a[n]=If[n==1,1,Sum[(-1)^(n-k-1)*a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,50]

A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
Wikipedia, Dirichlet convolution.

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).
From Antti Karttunen, Nov 22 2024: (Start)
Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
a(n) = Sum_{d|n} A323910(d).
a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
(End)

A342516 Number of strict integer partitions of n with weakly increasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 7, 8, 8, 11, 12, 14, 15, 17, 17, 21, 22, 26, 29, 31, 32, 35, 38, 42, 45, 48, 51, 58, 59, 63, 70, 76, 80, 88, 94, 98, 105, 113, 121, 129, 133, 143, 153, 159, 166, 183, 189, 195, 210, 221, 231, 248, 262, 273, 284, 298, 312

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also called log-concave-up strict partitions.
Also the number of reversed strict integer partitions of n with weakly increasing first quotients.
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 partition (6,3,2,1) has first quotients (1/2,2/3,1/2) so is not counted under a(12), even though the first differences (-3,-1,-1) are weakly increasing.
The a(1) = 1 through a(13) = 11 partitions (A..D = 10..13):
  1   2   3    4    5    6    7     8     9     A     B     C     D
          21   31   32   42   43    53    54    64    65    75    76
                    41   51   52    62    63    73    74    84    85
                              61    71    72    82    83    93    94
                              421   521   81    91    92    A2    A3
                                          621   532   A1    B1    B2
                                                721   632   732   C1
                                                      821   921   643
                                                                  832
                                                                  931
                                                                  A21

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 for differences instead of quotients is A179255.
The non-strict ordered version is A342492.
The non-strict version is A342497 (ranking: A342523).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with all adjacent parts x >= 2y.
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.
A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
Cf. A000005, A003114, A003242, A005117, A057567, A067824, A238710, A253249, A318991, A318992, A342528.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&LessEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342517 Number of strict integer partitions of n with strictly increasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 8, 10, 11, 13, 14, 16, 16, 19, 21, 23, 27, 29, 31, 34, 36, 40, 43, 47, 49, 53, 56, 59, 66, 71, 75, 81, 86, 89, 97, 104, 110, 119, 123, 132, 143, 148, 156, 168, 177, 184, 198, 209, 218, 232, 246, 257, 269, 282, 294

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also the number of reversed strict partitions of n with strictly increasing first quotients.

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 partition (14,8,5,3,2) has first quotients (4/7,5/8,3/5,2/3) so is not counted under a(32), even though the differences (-6,-3,-2,-1) are strictly increasing.
The a(1) = 1 through a(13) = 10 partitions (A..D = 10..13):
  1   2   3    4    5    6    7    8     9     A     B     C     D
          21   31   32   42   43   53    54    64    65    75    76
                    41   51   52   62    63    73    74    84    85
                              61   71    72    82    83    93    94
                                   521   81    91    92    A2    A3
                                         621   532   A1    B1    B2
                                               721   632   732   C1
                                                     821   921   643
                                                                 832
                                                                 A21

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 for differences instead of quotients is A179254.
The version for chains of divisors is A342086 (non-strict: A057567).
The non-strict ordered version is A342493.
The non-strict version is A342498 (ranking: A342524).
The weakly increasing version is A342516.
The strictly decreasing version is A342518.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Less@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342518 Number of strict integer partitions of n with strictly decreasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 8, 9, 11, 12, 13, 17, 18, 21, 24, 28, 30, 34, 37, 41, 47, 52, 56, 63, 68, 72, 83, 89, 99, 108, 117, 128, 139, 149, 163, 179, 189, 203, 217, 233, 250, 272, 289, 305, 329, 355, 381, 410, 438, 471, 505, 540, 571, 607, 645, 683, 726

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also the number of reversed strict integer partitions of n with strictly decreasing first quotients.

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 strict partition (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
The a(1) = 1 through a(13) = 12 partitions (A..D = 10..13):
  1   2   3    4    5    6     7    8     9     A      B     C     D
          21   31   32   42    43   53    54    64     65    75    76
                    41   51    52   62    63    73     74    84    85
                         321   61   71    72    82     83    93    94
                                    431   81    91     92    A2    A3
                                          432   541    A1    B1    B2
                                          531   631    542   543   C1
                                                4321   641   642   652
                                                       731   651   742
                                                             741   751
                                                             831   841
                                                                   5431

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 for differences instead of quotients is A320388.
The version for chains of divisors is A342086 (non-strict: A057567).
The non-strict ordered version is A342494.
The non-strict version is A342499 (ranking: A342525).
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A000041 counts partitions (strict: A000009).
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342096 counts partitions with all adjacent parts x < 2y (strict: A342097).
A342098 counts (strict) partitions with all adjacent parts x > 2y.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Greater@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342519 Number of strict integer partitions of n with weakly decreasing first quotients.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 12, 14, 15, 18, 18, 21, 25, 29, 32, 38, 40, 44, 51, 57, 61, 66, 73, 77, 89, 97, 104, 115, 124, 135, 147, 160, 174, 193, 206, 218, 238, 254, 272, 293, 313, 331, 353, 381, 408, 436, 468, 499, 532, 569, 610, 651, 694, 735, 783

Offset: 0

Gus Wiseman, Mar 20 2021

nonn

Also called log-concave-down strict partitions.
Also the number of reversed strict partitions of n with weakly decreasing first quotients.
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 strict partition (10,7,4,2,1) has first quotients (7/10,4/7,1/2,1/2) so is counted under a(24), even though the first differences (-3,-3,-2,-1) are weakly increasing.
The a(1) = 1 through a(13) = 14 strict partitions (A..D = 10..13):
  1   2   3    4    5    6     7     8     9     A      B     C      D
          21   31   32   42    43    53    54    64     65    75     76
                    41   51    52    62    63    73     74    84     85
                         321   61    71    72    82     83    93     94
                               421   431   81    91     92    A2     A3
                                           432   541    A1    B1     B2
                                           531   631    542   543    C1
                                                 4321   641   642    652
                                                        731   651    742
                                                              741    751
                                                              831    841
                                                              5421   931
                                                                     5431
                                                                     6421

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 non-strict ordered version is A069916.
The version for differences instead of quotients is A320382.
The non-strict version is A342513 (ranking: A342526).
The weakly increasing version is A342516.
The strictly decreasing version is A342518.
A000005 counts constant partitions.
A000041 counts partitions (strict: A000009).
A000929 counts partitions with all adjacent parts x >= 2y.
A001055 counts factorizations (strict: A045778, ordered: A074206).
A003238 counts chains of divisors summing to n - 1 (strict: A122651).
A057567 counts strict chains of divisors with weakly increasing quotients.
A167865 counts strict chains of divisors > 1 summing to n.
A342094 counts partitions with all adjacent parts x <= 2y (strict: A342095).
A342528 counts compositions with alternately weakly increasing parts.
Cf. A000005, A003114, A003242, A005117, A018819, A067824, A238710, A253249, A318991, A318992.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]

A342521 Heinz numbers of integer partitions with distinct first quotients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82

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 1365 are {2,3,4,6}, with first quotients (3/2,4/3,3/2), so 1365 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}
   24: {1,1,1,2}
   27: {2,2,2}
   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}
   54: {1,2,2,2}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   81: {2,2,2,2}
   84: {1,1,2,4}
   88: {1,1,1,5}
   96: {1,1,1,1,1,2}
  100: {1,1,3,3}

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

For multiplicities (prime signature) instead of quotients we have A130091.
For differences instead of quotients we have A325368 (count: A325325).
These partitions are counted by A342514 (strict: A342520, ordered: A342529).
The equal instead of distinct version is A342522.
The version counting strict divisor chains is A342530.
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.
Cf. A003242, A005117, A056239, A067824, A098859, A112798, A169594, A253249, A325326, A325337, A325405.

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

A343660 Number of maximal pairwise coprime sets of at least two divisors > 1 of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 4, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 8, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 6, 0, 1, 2, 2, 1, 4, 0, 4, 0, 1, 0, 8, 1, 1, 1

Offset: 1

Gus Wiseman, Apr 26 2021

nonn

			The a(n) sets for n = 6, 12, 24, 30, 36, 60, 72, 96:
  {2,3}  {2,3}  {2,3}  {5,6}    {2,3}  {5,6}    {2,3}  {2,3}
         {3,4}  {3,4}  {2,15}   {2,9}  {2,15}   {2,9}  {3,4}
                {3,8}  {3,10}   {3,4}  {3,10}   {3,4}  {3,8}
                       {2,3,5}  {4,9}  {3,20}   {3,8}  {3,16}
                                       {4,15}   {4,9}  {3,32}
                                       {5,12}   {8,9}
                                       {2,3,5}
                                       {3,4,5}

The case of pairs is A089233.
The case with 1's is A343652.
The case with singletons is (also) A343652.
The non-maximal version is A343653.
The non-maximal version with 1's is A343655.
The version for subsets of {2..n} is A343659 (for n > 2).
A018892 counts coprime unordered pairs of divisors.
A051026 counts pairwise indivisible subsets of {1..n}.
A066620 counts pairwise coprime 3-sets of divisors.
A100565 counts pairwise coprime unordered triples of divisors.
Cf. A005361, A007359, A007360, A067824, A074206, A225520, A276187, A320426, A325683, A326077, A326359, A326496, A337485, A343654.

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

a(n) = A343652(n) - A005361(n).

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.

cp's OEIS Frontend

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