A326843 - OEIS

%I #11 Jan 31 2021 02:58:55

%S 1,1,2,2,3,2,5,2,5,3,5,2,22,2,5,11,16,2,36,2,46,22,5,2,209,3,5,42,130,

%T 2,434,2,217,77,5,52,1400,2,5,135,1749,2,1782,2,957,2151,5,2,8355,3,

%U 1859,385,2388,2,6726,2765,10641,627,5,2,68049,2,5,13424,17142

%N Number of integer partitions of n whose length and maximum both divide n.

%C The Heinz numbers of these partitions are given by A326837.

%H Fausto A. C. Cariboni, <a href="/A326843/b326843.txt">Table of n, a(n) for n = 0..180</a>

%e The a(1) = 1 through a(8) = 5 partitions:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e        (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)

%e                     (1111)           (222)                (2222)

%e                                      (321)                (4211)

%e                                      (111111)             (11111111)

%e The a(12) = 22 partitions:

%e   (12)

%e   (6,6)

%e   (4,4,4)

%e   (6,3,3)

%e   (6,4,2)

%e   (6,5,1)

%e   (3,3,3,3)

%e   (4,3,3,2)

%e   (4,4,2,2)

%e   (4,4,3,1)

%e   (6,2,2,2)

%e   (6,3,2,1)

%e   (6,4,1,1)

%e   (2,2,2,2,2,2)

%e   (3,2,2,2,2,1)

%e   (3,3,2,2,1,1)

%e   (3,3,3,1,1,1)

%e   (4,2,2,2,1,1)

%e   (4,3,2,1,1,1)

%e   (4,4,1,1,1,1)

%e   (6,2,1,1,1,1)

%e   (1,1,1,1,1,1,1,1,1,1,1,1)

%t Table[If[n==0,1,Length[Select[IntegerPartitions[n],Divisible[n,Length[#]]&&Divisible[n,Max[#]]&]]],{n,0,30}]

%Y The strict case is A326851.

%Y The non-constant case is A326852.

%Y The case where all parts (not just the maximum) divide n is A326842.

%Y Cf. A018818, A047993, A067538, A326837, A326849.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 26 2019