A325186 Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 2.

3, 4, 15, 18, 21, 25, 27, 33, 36, 39, 51, 57, 69, 72, 87, 93, 105, 111, 123, 129, 141, 144, 147, 150, 159, 165, 175, 177, 183, 195, 201, 213, 219, 225, 231, 237, 245, 249, 250, 255, 267, 273, 275, 285, 288, 291, 300, 303, 309, 321, 325, 327, 339, 343, 345, 357

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 sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
   15: {2,3}
   18: {1,2,2}
   21: {2,4}
   25: {3,3}
   27: {2,2,2}
   33: {2,5}
   36: {1,1,2,2}
   39: {2,6}
   51: {2,7}
   57: {2,8}
   69: {2,9}
   72: {1,1,1,2,2}
   87: {2,10}
   93: {2,11}
  105: {2,3,4}
  111: {2,12}
  123: {2,13}
  129: {2,14}
		

Links

• Eric Weisstein's World of Mathematics, Graph Distance.
• Gus Wiseman, Young diagrams for the first 25 terms.

Crossrefs

Cf. A006918, A056239, A065770, A112798.

Cf. A325164, A325169, A325170, A325183, A325184, A325185, A325190, A325197.

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];
Select[Range[100],Apply[Plus,If[#==1,{},FixedPointList[corpos,ptnmat[primeptn[#]]][[-3]]],{0,1}]==2&]