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A326842 Number of integer partitions of n whose parts all divide n and whose length also divides n.

%I A326842 #8 Feb 20 2021 16:09:37
%S A326842 1,1,2,2,3,2,5,2,5,3,5,2,21,2,5,6,9,2,22,2,21,6,5,2,134,3,5,6,23,2,
%T A326842 157,2,27,6,5,6,478,2,5,6,208,2,224,2,31,63,5,2,1720,3,30,6,34,2,322,
%U A326842 6,295,6,5,2,13899,2,5,68,126,8,429,2,42,6,358,2,19959,2
%N A326842 Number of integer partitions of n whose parts all divide n and whose length also divides n.
%C A326842 The Heinz numbers of these partitions are given by A326847.
%H A326842 Fausto A. C. Cariboni, <a href="/A326842/b326842.txt">Table of n, a(n) for n = 0..419</a>
%e A326842 The a(1) = 1 through a(8) = 5 partitions:
%e A326842   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A326842        (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
%e A326842                     (1111)           (222)                (2222)
%e A326842                                      (321)                (4211)
%e A326842                                      (111111)             (11111111)
%e A326842 The a(12) = 21 partitions:
%e A326842   (12)
%e A326842   (6,6)
%e A326842   (4,4,4)
%e A326842   (6,3,3)
%e A326842   (6,4,2)
%e A326842   (3,3,3,3)
%e A326842   (4,3,3,2)
%e A326842   (4,4,2,2)
%e A326842   (4,4,3,1)
%e A326842   (6,2,2,2)
%e A326842   (6,3,2,1)
%e A326842   (6,4,1,1)
%e A326842   (2,2,2,2,2,2)
%e A326842   (3,2,2,2,2,1)
%e A326842   (3,3,2,2,1,1)
%e A326842   (3,3,3,1,1,1)
%e A326842   (4,2,2,2,1,1)
%e A326842   (4,3,2,1,1,1)
%e A326842   (4,4,1,1,1,1)
%e A326842   (6,2,1,1,1,1)
%e A326842   (1,1,1,1,1,1,1,1,1,1,1,1)
%t A326842 Table[Length[Select[IntegerPartitions[n,All,Divisors[n]],Divisible[n,Length[#]]&]],{n,1,30}]
%Y A326842 Partitions using divisors are A018818.
%Y A326842 Partitions whose length divides their sum are A067538.
%Y A326842 Cf. A047993, A102627, A316413, A326841, A326843, A326847.
%K A326842 nonn
%O A326842 0,3
%A A326842 _Gus Wiseman_, Jul 26 2019