A321649 Irregular triangle whose n-th row is the conjugate of the integer partition with Heinz number n.

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 2, 2, 2, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Nov 15 2018

Keywords

• nonn
• tabf

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			Triangle begins:
  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 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 1
  1 1 1 1 1 1 1 1 1
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).
		

Crossrefs

Cf. A008480, A056239, A112798, A122111, A296150, A321648, A321650.

Programs

Mathematica

primeMS[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[conj[primeMS[n]],{n,30}]

Formula

a(n,i) = A296150(A122111(n),i).