A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

1, 1, 1, 2, 2, 3, 2, 5, 5, 7, 7, 10, 10, 14, 14, 17, 19, 24, 24, 32, 33, 42, 43, 58, 59, 75, 79, 98, 104, 124, 128, 156, 166, 196, 204, 239, 251, 292, 306, 352, 372, 426, 445, 514, 543, 616, 652, 745, 790, 896, 960, 1080, 1162, 1311, 1400, 1574, 1692, 1892

Offset: 0

Author

Gus Wiseman, Jan 23 2021

Keywords

• nonn

Comments

The only strict partitions counted are (), (1), and (2,1).

Is there a simple generating function?

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  1  11  21   22    311    2211    331      2222      333
         111  1111  2111   111111  2221     4211      4221
                    11111          4111     221111    51111
                                   211111   311111    222111
                                   1111111  11111111  321111
                                                      21111111
                                                      111111111
		

Crossrefs

Note: Heinz numbers are given in parentheses below.

The reciprocal version is A143773 (A316428), with strict case A340830.

The case where length also divides n is A326842 (A326847).

The Heinz numbers of these partitions are A340606.

The version for factorizations is A340851, with reciprocal version A340853.

A018818 counts partitions of n into divisors of n (A326841).

A047993 counts balanced partitions (A106529).

A067538 counts partitions of n whose length/max divides n (A316413/A326836).

A067539 counts partitions with integer geometric mean (A326623).

A072233 counts partitions by sum and length.

A168659 = partitions whose greatest part divides their length (A340609).

A168659 = partitions whose length divides their greatest part (A340610).

A326843 = partitions of n whose length and maximum both divide n (A326837).

A330950 = partitions of n whose Heinz number is divisible by n (A324851).

Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]