A325195 Difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram of the integer partition with Heinz number n.

0, 0, 1, 1, 2, 0, 3, 2, 1, 1, 4, 1, 5, 2, 1, 3, 6, 1, 7, 1, 2, 3, 8, 2, 2, 4, 2, 2, 9, 0, 10, 4, 3, 5, 2, 2, 11, 6, 4, 2, 12, 1, 13, 3, 1, 7, 14, 3, 3, 1, 5, 4, 15, 2, 3, 2, 6, 8, 16, 1, 17, 9, 1, 5, 4, 2, 18, 5, 7, 1, 19, 3, 20, 10, 1, 6, 3, 3, 21, 3, 3, 11

Offset

1,5

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..82.

FindStat, St000384: The maximal part of the shifted composition of an integer partition

FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram

Example

The partition (3,3) has Heinz number 25 and diagram
  o o o
  o o o
containing maximal triangular partition
  o o
  o
and contained in minimal triangular partition
  o o o o
  o o o
  o o
  o
so a(25) = 4 - 2 = 2.

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&, Append[ptn, 0]];
Table[otbmax[primeptn[n]]-otb[primeptn[n]], {n, 100}]

Crossrefs

Cf. A046660, A065770, A071724, A243055, A325166, A325169, A325178, A325183, A325188, A325189, A325191, A325196, A325197, A325199, A325200.

Sequence in context: A352910 A362328 A342176 * A026728 A241556 A242029

Adjacent sequences: A325192 A325193 A325194 * A325196 A325197 A325198

Keyword

nonn

Author

Gus Wiseman, Apr 11 2019

Status

approved