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%I #8 Nov 15 2018 21:12:02
%S 1,1,1,2,1,1,1,2,1,1,1,1,1,3,2,2,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,2,1,
%T 1,1,2,2,1,4,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,3,1,1,2,2,1,1,2,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,4,1,2,2,2,2,1,1,1
%N Irregular triangle whose n-th row is the conjugate of the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%F a(n,i) = A296150(A122111(n),i).
%e Triangle begins:
%e 1
%e 1 1
%e 2
%e 1 1 1
%e 2 1
%e 1 1 1 1
%e 3
%e 2 2
%e 2 1 1
%e 1 1 1 1 1
%e 3 1
%e 1 1 1 1 1 1
%e 2 1 1 1
%e 2 2 1
%e 4
%e 1 1 1 1 1 1 1
%e 3 2
%e 1 1 1 1 1 1 1 1
%e 3 1 1
%e 2 2 1 1
%e 2 1 1 1 1
%e 1 1 1 1 1 1 1 1 1
%e The sequence of dual partitions begins: (), (1), (11), (2), (111), (21), (1111), (3), (22), (211), (11111), (31), (111111), (2111), (221), (4), (1111111), (32), (11111111), (311), (2211), (21111), (111111111), (41), (222), (211111), (33), (3111), (1111111111), (321).
%t primeMS[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[conj[primeMS[n]],{n,30}]
%Y Cf. A008480, A056239, A112798, A122111, A296150, A321648, A321650.
%K nonn,tabf
%O 1,4
%A _Gus Wiseman_, Nov 15 2018
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