A325163 Heinz number of the inner lining partition of the integer partition with Heinz number n.

1, 2, 3, 3, 5, 5, 7, 5, 10, 7, 11, 7, 13, 11, 14, 7, 17, 14, 19, 11, 22, 13, 23, 11, 21, 17, 21, 13, 29, 22, 31, 11, 26, 19, 33, 22, 37, 23, 34, 13, 41, 26, 43, 17, 33, 29, 47, 13, 55, 33, 38, 19, 53, 33, 39, 17, 46, 31, 59, 26, 61, 37, 39, 13, 51, 34, 67, 23

Offset: 1

Author

Gus Wiseman, Apr 05 2019

Keywords

• nonn

Comments

The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has diagonal distances
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining partition is (9,6,4), which has Heinz number 2093, so a(7865) = 2093.
		

Crossrefs

Cf. A052126, A056239, A064989, A065770, A093641, A112798, A188674, A252464, A257990, A297113, A325133, A325135, A325164, A325167, A325169.

Programs

Mathematica

Table[Times@@Prime/@(-Differences[Total/@Take[FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,Reverse[Flatten[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{1,-2}]]),{n,100}]