A363219 Twice the median of the conjugate of the integer partition with Heinz number n.

0, 2, 2, 4, 2, 3, 2, 6, 4, 2, 2, 4, 2, 2, 4, 8, 2, 5, 2, 2, 3, 2, 2, 5, 4, 2, 6, 2, 2, 4, 2, 10, 2, 2, 4, 6, 2, 2, 2, 2, 2, 3, 2, 2, 6, 2, 2, 6, 4, 4, 2, 2, 2, 7, 4, 2, 2, 2, 2, 4, 2, 2, 4, 12, 3, 2, 2, 2, 2, 4, 2, 7, 2, 2, 6, 2, 4, 2, 2, 2, 8, 2, 2, 3, 2, 2

Offset: 1

Author

Gus Wiseman, May 25 2023

Keywords

• nonn

Comments

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). Since the denominator is always 1 or 2, the median can be represented as an integer by multiplying by 2.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Examples

			The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.
		

Crossrefs

Twice the row media of A321649 or A321650.

For mean instead of twice median we have A326839/A326840.

This is the conjugate version of A360005.

A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).

A056239 adds up prime indices, row sums of A112798 and A296150.

A122111 is partition conjugation in terms of Heinz numbers.

A124010 gives prime signature, sorted A118914, length A001221, sum A001222.

A352491 gives n minus Heinz number of conjugate.

A363220 counts partitions with same median as conjugate.

Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]