A325183 Heinz number of the origin-to-boundary partition of the Young diagram of the integer partition with Heinz number n.

1, 2, 3, 3, 5, 6, 7, 5, 10, 10, 11, 10, 13, 14, 15, 7, 17, 15, 19, 14, 21, 22, 23, 14, 21, 26, 21, 22, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 22, 41, 42, 43, 26, 42, 46, 47, 22, 55, 42, 51, 34, 53, 35, 55, 26, 57, 58, 59, 42, 61, 62, 66, 13, 65, 66, 67

Offset

1,2

Comments

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary.

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..67.

Eric Weisstein's World of Mathematics, Graph Distance.

Example

The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with origin-to-boundary graph-distances
  4 4 4 3 2 1
  3 3 3 2 1
  2 2 2 1 1
  1 1 1
giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.

Mathematica

primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
ptnmat[ptn_]:=PadRight[(ConstantArray[1, #]&)/@Sort[ptn, Greater], {Length[ptn], Max@@ptn}+1];
corpos[mat_]:=ReplacePart[mat, Select[Position[mat, 1], Times@@Extract[mat, {#+{1, 0}, #+{0, 1}}]==0&]->0];
Table[Times@@Prime/@If[n==1, {}, -Differences[Map[Total, Drop[FixedPointList[corpos, ptnmat[primeptn[n]]], -1], 2]]], {n, 30}]

Crossrefs

The only terms appearing only once are the primorials A002110.

The union consists of all squarefree numbers A005117.

Cf. A000245, A056239, A065770, A112798, A174090, A297113.

Cf. A325166, A325167, A325169, A325184, A325188, A325189, A325195.

Sequence in context: A155918 A344705 A331170 * A343247 A097248 A097247

Adjacent sequences: A325180 A325181 A325182 * A325184 A325185 A325186

Keyword

nonn

Author

Gus Wiseman, Apr 08 2019

Status

approved