A340693 - OEIS

A340693

Number of integer partitions of n where each part is a divisor of the number of parts.

%I #11 Jan 25 2021 19:04:37

%S 1,1,1,2,2,3,2,5,5,7,7,10,10,14,14,17,19,24,24,32,33,42,43,58,59,75,

%T 79,98,104,124,128,156,166,196,204,239,251,292,306,352,372,426,445,

%U 514,543,616,652,745,790,896,960,1080,1162,1311,1400,1574,1692,1892

%N Number of integer partitions of n where each part is a divisor of the number of parts.

%C The only strict partitions counted are (), (1), and (2,1).

%C Is there a simple generating function?

%e The a(1) = 1 through a(9) = 7 partitions:

%e 1 11 21 22 311 2211 331 2222 333

%e 111 1111 2111 111111 2221 4211 4221

%e 11111 4111 221111 51111

%e 211111 311111 222111

%e 1111111 11111111 321111

%e 21111111

%e 111111111

%t Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]

%Y Note: Heinz numbers are given in parentheses below.

%Y The reciprocal version is A143773 (A316428), with strict case A340830.

%Y The case where length also divides n is A326842 (A326847).

%Y The Heinz numbers of these partitions are A340606.

%Y The version for factorizations is A340851, with reciprocal version A340853.

%Y A018818 counts partitions of n into divisors of n (A326841).

%Y A047993 counts balanced partitions (A106529).

%Y A067538 counts partitions of n whose length/max divides n (A316413/A326836).

%Y A067539 counts partitions with integer geometric mean (A326623).

%Y A072233 counts partitions by sum and length.

%Y A168659 = partitions whose greatest part divides their length (A340609).

%Y A168659 = partitions whose length divides their greatest part (A340610).

%Y A326843 = partitions of n whose length and maximum both divide n (A326837).

%Y A330950 = partitions of n whose Heinz number is divisible by n (A324851).

%Y Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 23 2021