A363219 - OEIS

%I #7 May 27 2023 16:45:20

%S 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,

%T 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,

%U 2,4,2,7,2,2,6,2,4,2,2,2,8,2,2,3,2,2

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

%C 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.

%C 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.

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

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]

%Y Twice the row media of A321649 or A321650.

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

%Y This is the conjugate version of A360005.

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

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

%Y A122111 is partition conjugation in terms of Heinz numbers.

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

%Y A352491 gives n minus Heinz number of conjugate.

%Y A363220 counts partitions with same median as conjugate.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, May 25 2023