A067731 -id:A067731 - cp's OEIS frontend

A067694 Minimum number of distinct parts in a self-conjugate partition of n, or 0 if n=2.

0, 1, 0, 2, 1, 2, 3, 2, 2, 1, 3, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 1, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 1, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 1, 2, 3

Offset: 0

Naohiro Nomoto, Feb 05 2002

There are no self-conjugate partitions of 2, so we set a(2)=0.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A000700, A067731.

a[0]=a[2]=0; a[n_] := Which[IntegerQ[Sqrt[n]], 1, Mod[n, 4]==2, 3, True, 2]

A067694(n) = if((2==n)||!n,0,if(2==(n%4),3,if(issquare(n),1,2))); \\ Antti Karttunen, Sep 27 2018

a(0)=a(2)=0; a(n^2)=1; a(4n+2)=3 for n>0; a(n)=2 in all other cases.

Edited by Dean Hickerson, Feb 15 2002

A240450 Greatest number of distinct numbers in the intersection of p and its conjugate, as p ranges through the partitions of n.

Original entry on oeis.org

2, 1, 3, 2, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 9, 8, 9, 8, 9

Offset: 1

Clark Kimberling, Apr 12 2014

Number of terms in row n of the array at A240181.

To match the definition, all terms need to be decreased by 1 (because the rows in A240181 start with k=0). So this appears to be an incorrect duplicate of A067731. - Joerg Arndt, Jul 30 2017

Cf. A067731, A240181.

z = 30; conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; c = Map[BinCounts[#, {0, 1 + Max[#]}] &[Map[Length, Map[Intersection[#, conjugatePartition[#]] &, IntegerPartitions[#]]]] &, Range[z]]; Flatten[c]  (* A240181 *)
Table[Length[c[[n]]], {n, 1, z}] (* A240450 *) (* Peter J. C. Moses, Apr 10 2014 *)

cp's OEIS Frontend

A067694 Minimum number of distinct parts in a self-conjugate partition of n, or 0 if n=2.

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