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Number of strict integer partitions of n such that every part is a multiple of the number of parts.
11

%I #8 Feb 03 2021 09:09:08

%S 1,1,1,1,1,2,1,2,1,3,1,3,1,4,1,4,1,6,1,5,2,6,1,8,1,7,4,7,1,12,1,8,6,9,

%T 1,16,1,10,9,11,1,21,1,12,13,12,1,28,1,13,17,16,1,33,1,19,22,15,1,45,

%U 1,16,28,25,1,47,1,28,34,18

%N Number of strict integer partitions of n such that every part is a multiple of the number of parts.

%F a(n) = Sum_{d|n} A008289(n/d, d).

%e The a(n) partitions for n = 1, 6, 10, 14, 18, 20, 24, 26, 30:

%e 1 6 10 14 18 20 24 26 30

%e 4,2 6,4 8,6 10,8 12,8 16,8 18,8 22,8

%e 8,2 10,4 12,6 14,6 18,6 20,6 24,6

%e 12,2 14,4 16,4 20,4 22,4 26,4

%e 16,2 18,2 22,2 24,2 28,2

%e 9,6,3 14,10 14,12 16,14

%e 12,9,3 16,10 18,12

%e 15,6,3 20,10

%e 15,9,6

%e 18,9,3

%e 21,6,3

%e 15,12,3

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@IntegerQ/@(#/Length[#])&]],{n,30}]

%Y Note: A-numbers of Heinz-number sequences are in parentheses below.

%Y The non-strict case is A143773 (A316428).

%Y The case where length divides sum also is A340827.

%Y The version for factorizations is A340851.

%Y Factorization of this type are counted by A340853.

%Y A018818 counts partitions into divisors (A326841).

%Y A047993 counts balanced partitions (A106529).

%Y A067538 counts partitions whose length/max divide sum (A316413/A326836).

%Y A072233 counts partitions by sum and length, with strict case A008289.

%Y A102627 counts strict partitions whose length divides sum.

%Y A326850 counts strict partitions whose maximum part divides sum.

%Y A326851 counts strict partitions with length and maximum dividing sum.

%Y A340828 counts strict partitions with length divisible by maximum.

%Y A340829 counts strict partitions with Heinz number divisible by sum.

%Y Cf. A114638, A168659, A326641, A326843 (A326837), A326849, A326852 (A326838), A330950 (A324851), A340852.

%K nonn

%O 1,6

%A _Gus Wiseman_, Feb 02 2021