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

1,1

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 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).

Links

Table of n, a(n) for n=1..62.

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:
    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}

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]]&]

Crossrefs

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

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

Sequence in context: A108370 A325706 A367683 * A285532 A350944 A016726

Adjacent sequences: A325182 A325183 A325184 * A325186 A325187 A325188

Keyword

nonn

Author

Gus Wiseman, Apr 08 2019

Status

approved