A325183 - OEIS

%I #5 Apr 10 2019 22:02:09

%S 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,

%T 21,22,29,30,31,11,33,34,35,21,37,38,39,22,41,42,43,26,42,46,47,22,55,

%U 42,51,34,53,35,55,26,57,58,59,42,61,62,66,13,65,66,67

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

%C 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.

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphDistance.html">Graph Distance</a>.

%e The partition with Heinz number 7865 is (6,5,5,3), with diagram

%e   o o o o o o

%e   o o o o o

%e   o o o o o

%e   o o o

%e with origin-to-boundary graph-distances

%e   4 4 4 3 2 1

%e   3 3 3 2 1

%e   2 2 2 1 1

%e   1 1 1

%e giving the origin-to-boundary partition (7,5,4,3) with Heinz number 6545, so a(7865) = 6545.

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t ptnmat[ptn_]:=PadRight[(ConstantArray[1,#]&)/@Sort[ptn,Greater],{Length[ptn],Max@@ptn}+1];

%t corpos[mat_]:=ReplacePart[mat,Select[Position[mat,1],Times@@Extract[mat,{#+{1,0},#+{0,1}}]==0&]->0];

%t Table[Times@@Prime/@If[n==1,{},-Differences[Map[Total,Drop[FixedPointList[corpos,ptnmat[primeptn[n]]],-1],2]]],{n,30}]

%Y The only terms appearing only once are the primorials A002110.

%Y The union consists of all squarefree numbers A005117.

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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 08 2019