cp's OEIS Frontend

A340693 Number of integer partitions of n where each part is a divisor of the number of parts.

%I A340693 #11 Jan 25 2021 19:04:37
%S A340693 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 A340693 79,98,104,124,128,156,166,196,204,239,251,292,306,352,372,426,445,
%U A340693 514,543,616,652,745,790,896,960,1080,1162,1311,1400,1574,1692,1892
%N A340693 Number of integer partitions of n where each part is a divisor of the number of parts.
%C A340693 The only strict partitions counted are (), (1), and (2,1).
%C A340693 Is there a simple generating function?
%e A340693 The a(1) = 1 through a(9) = 7 partitions:
%e A340693   1  11  21   22    311    2211    331      2222      333
%e A340693          111  1111  2111   111111  2221     4211      4221
%e A340693                     11111          4111     221111    51111
%e A340693                                    211111   311111    222111
%e A340693                                    1111111  11111111  321111
%e A340693                                                       21111111
%e A340693                                                       111111111
%t A340693 Table[Length[Select[IntegerPartitions[n],And@@IntegerQ/@(Length[#]/#)&]],{n,0,30}]
%Y A340693 Note: Heinz numbers are given in parentheses below.
%Y A340693 The reciprocal version is A143773 (A316428), with strict case A340830.
%Y A340693 The case where length also divides n is A326842 (A326847).
%Y A340693 The Heinz numbers of these partitions are A340606.
%Y A340693 The version for factorizations is A340851, with reciprocal version A340853.
%Y A340693 A018818 counts partitions of n into divisors of n (A326841).
%Y A340693 A047993 counts balanced partitions (A106529).
%Y A340693 A067538 counts partitions of n whose length/max divides n (A316413/A326836).
%Y A340693 A067539 counts partitions with integer geometric mean (A326623).
%Y A340693 A072233 counts partitions by sum and length.
%Y A340693 A168659 = partitions whose greatest part divides their length (A340609).
%Y A340693 A168659 = partitions whose length divides their greatest part (A340610).
%Y A340693 A326843 = partitions of n whose length and maximum both divide n (A326837).
%Y A340693 A330950 = partitions of n whose Heinz number is divisible by n (A324851).
%Y A340693 Cf. A000041, A003114, A006141, A033630, A064174, A074761, A102627, A200750, A237984, A298423, A340827.
%K A340693 nonn
%O A340693 0,4
%A A340693 _Gus Wiseman_, Jan 23 2021