A325196 Heinz numbers of integer partitions such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is 1.

3, 4, 9, 10, 12, 15, 18, 20, 42, 45, 50, 60, 63, 70, 75, 84, 90, 100, 105, 126, 140, 150, 294, 315, 330, 350, 420, 441, 462, 490, 495, 525, 550, 588, 630, 660, 693, 700, 735, 770, 825, 882, 924, 980, 990, 1050, 1100, 1155, 1386, 1470, 1540, 1650, 2730, 3234

Offset: 1

Author

Gus Wiseman, Apr 11 2019

Keywords

• nonn

Comments

The enumeration of these partitions by sum is given by A325191.

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}
    4: {1,1}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   42: {1,2,4}
   45: {2,2,3}
   50: {1,3,3}
   60: {1,1,2,3}
   63: {2,2,4}
   70: {1,3,4}
   75: {2,3,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
  105: {2,3,4}
  126: {1,2,2,4}
		

Links

• Gus Wiseman, Young diagrams for their first 60 terms.
• FindStat, St000783: The maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram
• FindStat, St000384: The maximal part of the shifted composition of an integer partition.

Crossrefs

Cf. A060687, A065770, A256617, A325166, A325169, A325179, A325181, A325183, A325185, A325188, A325189, A325191, A325195, A325200.

Programs

Mathematica

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
otbmax[ptn_]:=Max@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Select[Range[1000],otbmax[primeptn[#]]-otb[primeptn[#]]==1&]