This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A217605 #28 May 14 2019 12:25:35
%S A217605 1,1,0,0,2,1,1,0,1,1,3,0,3,3,3,0,4,3,2,1,6,4,5,2,5,7,10,2,10,10,11,4,
%T A217605 9,5,14,7,13,13,18,7,20,17,22,10,22,19,32,15,26,26,40,15,37,36,43,21,
%U A217605 44,32,55,30,46,43,75,32,67,62,83,40,82,61,104,58,89,71,136,66,114,97,149,77,143,106,176,101,160,123,222,114,190
%N A217605 Number of partitions of n that are fixed points of a certain map (see comment).
%C A217605 Writing a partition of n in the form sum(k>=1, c(k) * k) another (in general different) partition is obtained as sum(k>=1, k * c(k)). For example, the partition 6 = 4* 1 + 1* 2 = 1 + 1 + 1 + 1 + 2 is mapped to 1* 4 + 2 *1 = 2* 1 + 1* 4 = 2 + 2 + 4. This sequence counts the fixed points of this map.
%C A217605 The map is not surjective. For example, all partitions into distinct parts are mapped to n* 1.
%C A217605 The map is an involution for partitions where the multiplicities of all parts are distinct (Wilf partitions, see A098859). If in addition the set of parts the same as the set of multiplicities then the partition is a fixed point.
%C A217605 The second part of the preceding comment is incorrect. For example, the partition (3,3,2,1,1,1) maps to (3,2,2,2,1,1) so is not a fixed point, even though the set of parts is identical to the set of multiplicities. - _Gus Wiseman_, May 04 2019
%H A217605 James Allen Fill, Svante Janson, Mark Daniel Ward, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p18">Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf</a>, The Electronic Journal of Combinatorics, vol.19, no.2, 2012.
%H A217605 Stephan Wagner, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p13">The Number of Fixed Points of Wilf's Partition Involution</a>, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
%e A217605 a(16) = 4 because the following partitions of 16 are fixed points:
%e A217605 4* 2 + 2* 4 = 2 + 2 + 2 + 2 + 4 + 4
%e A217605 4* 4 = 4 + 4 + 4 + 4
%e A217605 6* 1 + 2* 2 + 1* 6 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 6
%e A217605 8* 1 + 1* 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 8
%e A217605 From _Gus Wiseman_, May 04 2019: (Start)
%e A217605 The a(1) = 1 through a(16) = 4 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A048768.
%e A217605 1 22 221 3111 41111 333 3331 33222 33322 333221 4444
%e A217605 211 322111 4221111 332221 52211111 442222
%e A217605 511111 6111111 333211 71111111 622111111
%e A217605 811111111
%e A217605 (End)
%t A217605 winv[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
%t A217605 Table[Length[Select[IntegerPartitions[n],winv[Times@@Prime/@#]==Times@@Prime/@#&]],{n,0,30}] (* _Gus Wiseman_, May 04 2019 *)
%Y A217605 Cf. A033461, A048767, A048768, A098859, A320348, A325324, A325325.
%K A217605 nonn
%O A217605 0,5
%A A217605 _Joerg Arndt_, Oct 08 2012