

A330953


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


18



1, 2, 1, 2, 1, 3, 3, 4, 6, 3, 12, 10, 12, 14, 27, 38, 44, 52, 48, 77, 101, 106, 127, 206, 268, 377, 392, 496, 602, 671, 821, 1090, 1318, 1568, 1926, 2260, 2703, 3258, 3942, 4858, 5923, 6891, 8286, 9728, 11676, 13775, 16314, 19749, 23474, 27793, 32989, 38775
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OFFSET

1,2


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..52.


EXAMPLE

The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
1 2 3 4 5 6 7 8 9 A B
11 1111 222 3211 431 432 5311 542
321 22111 4211 3321 22111111 5411
11111111 32211 33221
321111 42221
2211111 53111
322211
431111
521111
2222111
3311111
32111111
For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.


MATHEMATICA

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


CROSSREFS

The Heinz numbers of these partitions are given by A036844.
Numbers divisible by the sum of their prime indices are 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.
Partitions whose Heinz number is divisible by their sum are A330950.
Partitions whose product is divisible by their sum of primes are A330954.
Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.
Sequence in context: A029174 A058753 A282249 * A325702 A304746 A304748
Adjacent sequences: A330950 A330951 A330952 * A330954 A330955 A330956


KEYWORD

nonn


AUTHOR

Gus Wiseman, Jan 15 2020


STATUS

approved



