A330952 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by all parts.

1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 14, 20, 25, 32, 42, 54, 69, 87, 109, 137, 172, 215, 269, 331, 409, 499, 612, 751, 917, 1111, 1344, 1626, 1963, 2359, 2834, 3396, 4065, 4849, 5779, 6865, 8146, 9658, 11424, 13483, 15898, 18710, 21999, 25823, 30272, 35417, 41397

Offset: 0

Author

Gus Wiseman, Jan 15 2020

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The a(1) = 1 through a(9) = 11 partitions:
  1  11  21   211   221    321     2221     3221      621
         111  1111  2111   411     3211     4211      3321
                    11111  2211    4111     22211     22221
                           21111   22111    32111     32211
                           111111  211111   41111     42111
                                   1111111  221111    222111
                                            2111111   321111
                                            11111111  411111
                                                      2211111
                                                      21111111
                                                      111111111
		

Crossrefs

The Heinz numbers of these partitions are given by A120383.

Partitions whose product is divisible by their sum are A057568.

Partitions whose Heinz number is divisible by their product are A324925.

Partitions whose Heinz number is divisible by their sum are A330950.

Cf. A056239, A112798, A324756, A326149, A326155, A330953, A330954, A331383.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@Table[Divisible[Times@@Prime/@#,i],{i,#}]&]],{n,0,30}]