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

1,1

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).

Links

Table of n, a(n) for n=1..63.

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

Example

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}

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&]

Crossrefs

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

Sequence in context: A091886 A333357 A111774 * A325229 A036347 A129492

Adjacent sequences: A325167 A325168 A325169 * A325171 A325172 A325173

Keyword

nonn

Author

Gus Wiseman, Apr 05 2019

Status

approved