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

Author

Gus Wiseman, Mar 25 2021

Keywords

• nonn

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).

Examples

			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.
		

Links

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

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.

Programs

Mathematica

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

Formula

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