cp's OEIS Frontend

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

%I A363219 #7 May 27 2023 16:45:20
%S A363219 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 A363219 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 A363219 2,4,2,7,2,2,6,2,4,2,2,2,8,2,2,3,2,2
%N A363219 Twice the median of the conjugate of the integer partition with Heinz number n.
%C A363219 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 A363219 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 A363219 The partition (4,2,1) has Heinz number 42 and conjugate (3,2,1,1) with median 3/2, so a(42) = 3.
%t A363219 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363219 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A363219 Table[If[n==1,0,2*Median[conj[prix[n]]]],{n,100}]
%Y A363219 Twice the row media of A321649 or A321650.
%Y A363219 For mean instead of twice median we have A326839/A326840.
%Y A363219 This is the conjugate version of A360005.
%Y A363219 A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
%Y A363219 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A363219 A122111 is partition conjugation in terms of Heinz numbers.
%Y A363219 A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
%Y A363219 A352491 gives n minus Heinz number of conjugate.
%Y A363219 A363220 counts partitions with same median as conjugate.
%Y A363219 Cf. A046682, A321648, A325040, A326567/A326568, A326848, A362617, A362618, A363223, A363224.
%K A363219 nonn
%O A363219 1,2
%A A363219 _Gus Wiseman_, May 25 2023