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

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

1,2

Comments

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).

Links

Table of n, a(n) for n=1..78.

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

Example

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.

Mathematica

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

Crossrefs

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.

Sequence in context: A197929 A326849 A328706 * A240090 A078224 A159688

Adjacent sequences: A342527 A342528 A342529 * A342531 A342532 A342533

Keyword

nonn

Author

Gus Wiseman, Mar 25 2021

Status

approved