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

Author

Gus Wiseman, Apr 08 2019

Keywords

• nonn

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).

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Graph Distance.

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.

Programs

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}]