A330950 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

1, 1, 1, 2, 2, 3, 3, 7, 7, 11, 11, 22, 15, 30, 42, 77, 42, 101, 56, 176, 176, 231, 135, 490, 490, 490, 792, 1002, 490, 1575, 627, 3010, 2436, 2436, 3718, 5604, 1958, 4565, 6842, 12310, 3718, 14883, 4565, 21637, 26015, 17977, 8349, 53174, 44583, 63261

Offset

1,4

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.

Links

Table of n, a(n) for n=1..50.

Example

The a(1) = 1 through a(10) = 11 partitions:
  1  11  21  211   32   321    43    5111      522      631
             1111  311  2211   421   32111     3222     3331
                        21111  4111  41111     4221     4321
                                     221111    22221    5311
                                     311111    32211    32221
                                     2111111   222111   33211
                                     11111111  2211111  43111
                                                        322111
                                                        331111
                                                        3211111
                                                        31111111
For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).

Mathematica

Table[Length[Select[IntegerPartitions[n], Divisible[Times@@Prime/@#, n]&]], {n, 20}]

Crossrefs

The Heinz numbers of these partitions are given by A324851.

Partitions whose product is divisible by their sum are A057568.

Partitions whose Heinz number is divisible by all parts are A330952.

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

Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.

Sequence in context: A324520 A307737 A309684 * A032060 A241385 A307736

Adjacent sequences: A330947 A330948 A330949 * A330951 A330952 A330953

Keyword

nonn

Author

Gus Wiseman, Jan 15 2020

Status

approved