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

0,4

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).

Links

Table of n, a(n) for n=0..69.

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.

Example

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

Mathematica

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

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.

Sequence in context: A332893 A113605 A369984 * A307815 A070230 A007150

Adjacent sequences: A342512 A342513 A342514 * A342516 A342517 A342518

Keyword

nonn

Author

Gus Wiseman, Mar 19 2021

Status

approved