A330952 - OEIS

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

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OFFSET

0,4

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=0..50.

EXAMPLE

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

MATHEMATICA

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

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.

Sequence in context: A308858 A192432 A121081 * A118399 A278298 A178927

Adjacent sequences: A330949 A330950 A330951 * A330953 A330954 A330955

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 15 2020

STATUS

approved