A342496 Number of integer partitions of n with constant (equal) first quotients.

1, 1, 2, 3, 4, 4, 6, 6, 7, 7, 8, 7, 11, 9, 11, 12, 12, 10, 14, 12, 15, 16, 14, 13, 19, 15, 17, 17, 20, 16, 23, 19, 21, 20, 20, 22, 26, 21, 23, 25, 28, 22, 30, 24, 27, 29, 26, 25, 33, 29, 30, 29, 32, 28, 34, 31, 36, 34, 32, 31, 42

Offset: 0

Author

Gus Wiseman, Mar 17 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 first quotients of (6,3,1) are (1/2,1/3).

Examples

			The partition (12,6,3) has first quotients (1/2,1/2) so is counted under a(21).
The a(1) = 1 through a(9) = 7 partitions:
  1   2    3     4      5       6        7         8          9
      11   21    22     32      33       43        44         54
           111   31     41      42       52        53         63
                 1111   11111   51       61        62         72
                                222      421       71         81
                                111111   1111111   2222       333
                                                   11111111   111111111
		

Links

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

Crossrefs

The version for differences instead of quotients is A049988.

The ordered version is A342495.

The distinct version is A342514.

The strict case is A342515.

The Heinz numbers of these partitions are A342522.

A000005 counts constant partitions.

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

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

Cf. A000837, A002843, A003242, A074206, A175342, A318991, A318992, A325557, A342527, A342528, A342529.

Programs

Mathematica

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

Formula

a(n > 0) = (A342495(n) + A000005(n))/2.