%I #10 Feb 08 2021 03:05:59

%S 1,1,2,1,2,3,3,2,4,5,6,6,7,8,11,10,13,17,18,21,24,27,30,35,39,46,53,

%T 61,68,79,87,97,110,123,139,157,175,196,222,247,278,312,347,385,433,

%U 476,531,586,651,720,800,883,979,1085,1200,1325,1464,1614,1777

%N Number of strict integer partitions of n whose maximum part is a multiple of their length.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000010">St000010: The length of the partition.</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000147">St000147: The largest part of an integer partition.</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000784">St000784: The maximum of the length and the largest part of the integer partition.</a>

%e The a(1) = 1 through a(16) = 10 partitions (A..G = 10..16):

%e 1 2 3 4 5 6 7 8 9 A B C D E F G

%e 21 41 42 43 62 63 64 65 84 85 86 87 A6

%e 321 61 81 82 83 A2 A3 A4 A5 C4

%e 621 631 A1 642 C1 C2 C3 E2

%e 4321 632 651 643 653 E1 943

%e 641 921 652 932 654 952

%e 931 941 942 961

%e 8321 951 C31

%e C21 8431

%e 8421 8521

%e 54321

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Max@@#,Length[#]]&]],{n,30}]

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

%Y The non-strict version is A168659 (A340609/A340610).

%Y A018818 counts partitions into divisors (A326841).

%Y A047993 counts balanced partitions (A106529).

%Y A064173 counts partitions of positive/negative rank (A340787/A340788).

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

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

%Y A096401 counts strict partition with length equal to minimum.

%Y A102627 counts strict partitions with length dividing sum.

%Y A326842 counts partitions whose length and parts all divide sum (A326847).

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

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

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

%Y A340830 counts strict partitions with all parts divisible by length.

%Y Cf. A003114, A006141, A064174, A117409, A143773 (A316428), A200750, A326843 (A326837), A330950 (A324851).

%K nonn

%O 1,3

%A _Gus Wiseman_, Feb 01 2021