A325185 Heinz numbers of integer partitions such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

2, 6, 9, 10, 12, 14, 20, 22, 24, 26, 28, 30, 34, 38, 40, 42, 44, 45, 46, 48, 50, 52, 56, 58, 60, 62, 63, 66, 68, 70, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 110, 112, 114, 116, 117, 118, 120, 122, 124, 125, 126, 130, 132

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 sequence gives all Heinz numbers of integer partitions whose Young diagram has last part of its origin-to-boundary partition equal to 1.

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:
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   20: {1,1,3}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   34: {1,7}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   44: {1,1,5}
   45: {2,2,3}
   46: {1,9}
   48: {1,1,1,1,2}
		

Links

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

Crossrefs

Cf. A001222, A056239, A061395, A065770, A112798, A188674.

Cf. A325169, A325183, A325184, A325186, A325187, A325196.

Programs

Mathematica

hptn[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]];
Select[Range[2,100],otb[hptn[#]]>otb[Rest[hptn[#]]]&&otb[hptn[#]]>otb[DeleteCases[hptn[#]-1,0]]&]