A342515 Number of strict partitions of n with constant (equal) first-quotients.

1, 1, 1, 2, 2, 3, 3, 5, 4, 5, 5, 6, 6, 8, 8, 9, 8, 9, 9, 11, 10, 13, 11, 12, 12, 13, 14, 14, 15, 15, 16, 18, 16, 17, 17, 19, 18, 20, 20, 22, 21, 21, 23, 23, 22, 24, 23, 24, 24, 27, 25, 26, 27, 27, 27, 28, 29, 31, 29, 30, 31, 32, 33, 35, 32, 35, 33, 35, 34, 35

Offset: 0

Author

Gus Wiseman, Mar 19 2021

Keywords

• nonn

Comments

Also the number of reversed strict partitions of n with constant (equal) first-quotients.

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(15) = 9 partitions (A..F = 10..15):
  1   2   3    4    5    6    7     8    9    A    B    C    D     E     F
          21   31   32   42   43    53   54   64   65   75   76    86    87
                    41   51   52    62   63   73   74   84   85    95    96
                              61    71   72   82   83   93   94    A4    A5
                              421        81   91   92   A2   A3    B3    B4
                                                   A1   B1   B2    C2    C3
                                                             C1    D1    D2
                                                             931   842   E1
                                                                         8421
		

Links

• Wikipedia, Arithmetic progression
• 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.

Crossrefs

The version for differences instead of quotients is A049980.

The non-strict ordered version is A342495.

The non-strict version is A342496.

The distinct instead of equal version is A342520.

A000005 counts constant partitions.

A000041 counts partitions (strict: A000009).

A001055 counts factorizations (strict: A045778, ordered: A074206).

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A154402 counts partitions with adjacent parts x = 2y.

A167865 counts strict chains of divisors > 1 summing to n.

A175342 counts compositions with equal differences.

Cf. A003242, A005117, A049988, A057567, A067824, A253249, A307824, A318991, A318992, A325328, A342086.

Programs

Mathematica

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