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

1,1

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).

Links

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Graph Distance.

Gus Wiseman, Young diagrams for the first 25 terms.

Example

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}

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&]

Crossrefs

Cf. A006918, A056239, A065770, A112798.

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

Sequence in context: A341779 A136641 A373262 * A053359 A056742 A338123

Adjacent sequences: A325183 A325184 A325185 * A325187 A325188 A325189

Keyword

nonn

Author

Gus Wiseman, Apr 08 2019

Status

approved