A359358 Let y be the integer partition with Heinz number n. Then a(n) is the size of the Young diagram of y after removing a rectangle of the same length as y and width equal to the smallest part of y.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 2, 4, 0, 1, 0, 5, 0, 3, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 2, 0, 4, 0, 4, 1, 8, 0, 1, 0, 4, 5, 5, 0, 3, 2, 3, 6, 9, 0, 3, 0, 10, 2, 0, 3, 5, 0, 6, 7, 5, 0, 2, 0, 11, 2, 7, 1, 6, 0, 2, 0, 12, 0, 4, 4, 13

Offset: 1

Author

Gus Wiseman, Dec 27 2022

Keywords

• nonn

Comments

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), which has the following diagram. The 3 X 4 rectangle is shown in dots.
  . . . o o o
  . . . o o
  . . . o o
  . . .
The size of the complement is 7, so a(7865) = 7.
		

Crossrefs

The opposite version is A326844.

Row sums of A356958 are a(n) + A001222(n) - 1, Heinz numbers A246277.

A055396 gives minimum prime index, maximum A061395.

A112798 list prime indices, sum A056239.

A243055 subtracts the least prime index from the greatest.

A326846 = size of the smallest rectangle containing the prime indices of n.

A358195 gives Heinz numbers of rows of A358172, even bisection A241916.

Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

Programs

Mathematica

Table[If[n==1,0,With[{q=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Total[q]-q[[1]]*Length[q]]],{n,100}]

Formula

a(n) = A056239(n) - A001222(n) * A055396(n).

a(n) = A056239(n) - A359360(n).