A325170 Heinz numbers of integer partitions with origin-to-boundary graph-distance equal to 2.

6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 27, 28, 33, 34, 35, 36, 38, 39, 40, 44, 46, 48, 49, 51, 52, 54, 55, 56, 57, 58, 62, 65, 68, 69, 72, 74, 76, 77, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 104, 106, 108, 111, 112, 115, 116, 118, 119

Offset: 1

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

Comments

The origin-to-boundary graph-distance of a Young diagram is the minimum number of unit steps East or South from the upper-left square to a nonsquare in the lower-right quadrant. It is also the side-length of the minimum triangular partition contained inside the diagram.

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:
   6: {1,2}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}
  38: {1,8}
		

Links

• Gus Wiseman, Young diagrams corresponding to the first 50 terms.

Crossrefs

Cf. A001221, A001222, A006918, A056239, A065770, A112798, A174090, A257990, A297113, A325167, A325169.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[200],otb[Reverse[primeMS[#]]]==2&]