A340830 Number of strict integer partitions of n such that every part is a multiple of the number of parts.

1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 1, 4, 1, 6, 1, 5, 2, 6, 1, 8, 1, 7, 4, 7, 1, 12, 1, 8, 6, 9, 1, 16, 1, 10, 9, 11, 1, 21, 1, 12, 13, 12, 1, 28, 1, 13, 17, 16, 1, 33, 1, 19, 22, 15, 1, 45, 1, 16, 28, 25, 1, 47, 1, 28, 34, 18

Offset: 1

Author

Gus Wiseman, Feb 02 2021

Keywords

• nonn

Examples

			The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:
  1   6     10    14     18      20     24       26      30
      4,2   6,4   8,6    10,8    12,8   16,8     18,8    22,8
            8,2   10,4   12,6    14,6   18,6     20,6    24,6
                  12,2   14,4    16,4   20,4     22,4    26,4
                         16,2    18,2   22,2     24,2    28,2
                         9,6,3          14,10    14,12   16,14
                                        12,9,3   16,10   18,12
                                        15,6,3           20,10
                                                         15,9,6
                                                         18,9,3
                                                         21,6,3
                                                         15,12,3
		

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

The non-strict case is A143773 (A316428).

The case where length divides sum also is A340827.

The version for factorizations is A340851.

Factorization of this type are counted by A340853.

A018818 counts partitions into divisors (A326841).

A047993 counts balanced partitions (A106529).

A067538 counts partitions whose length/max divide sum (A316413/A326836).

A072233 counts partitions by sum and length, with strict case A008289.

A102627 counts strict partitions whose length divides sum.

A326850 counts strict partitions whose maximum part divides sum.

A326851 counts strict partitions with length and maximum dividing sum.

A340828 counts strict partitions with length divisible by maximum.

A340829 counts strict partitions with Heinz number divisible by sum.

Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]],{n,30}]

Formula

a(n) = Sum_{d|n} A008289(n/d, d).