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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 26 01:25 EDT 2020. Contains 337346 sequences. (Running on oeis4.)