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

1,4

Comments

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

Links

Table of n, a(n) for n=1..87.

Formula

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

Example

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).

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}]

Crossrefs

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

Sequence in context: A216784 A256067 A256554 * A003650 A059233 A357327

Adjacent sequences: A321646 A321647 A321648 * A321650 A321651 A321652

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 15 2018

Status

approved