A321650 Irregular triangle whose n-th row is the reversed conjugate of the integer partition with Heinz number n.

1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 2, 2, 2, 1, 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) = A112798(A122111(n),i).

Example

Triangle begins:
  1
  1 1
  2
  1 1 1
  1 2
  1 1 1 1
  3
  2 2
  1 1 2
  1 1 1 1 1
  1 3
  1 1 1 1 1 1
  1 1 1 2
  1 2 2
  4
  1 1 1 1 1 1 1
  2 3
  1 1 1 1 1 1 1 1
  1 1 3
  1 1 2 2
  1 1 1 1 2
  1 1 1 1 1 1 1 1 1
The sequence of reversed dual partitions begins: (), (1), (11), (2), (111), (12), (1111), (3), (22), (112), (11111), (13), (111111), (1112), (122), (4), (1111111), (23), (11111111), (113), (1122), (11112), (111111111), (14), (222), (111112), (33), (1113), (1111111111), (123).

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[Sort[conj[primeMS[n]]], {n, 50}]

Crossrefs

Cf. A008480, A056239, A112798, A122111, A296150, A321648, A321649.

Sequence in context: A247090 A030346 A030336 * A205007 A078470 A230799

Adjacent sequences: A321647 A321648 A321649 * A321651 A321652 A321653

Keyword

nonn,tabf

Author

Gus Wiseman, Nov 15 2018

Status

approved