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

1,10

Comments

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..86.

Formula

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

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

Example

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.

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}]

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.

Sequence in context: A242411 A286470 A243055 * A318371 A363157 A363952

Adjacent sequences: A359355 A359356 A359357 * A359359 A359360 A359361

Keyword

nonn

Author

Gus Wiseman, Dec 27 2022

Status

approved