A326844 Let y be the integer partition with Heinz number n. Then a(n) is the size of the complement, in the minimal rectangular partition containing the Young diagram of y, of the Young diagram of y.

0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 2, 0, 3, 1, 0, 0, 1, 0, 4, 2, 4, 0, 3, 0, 5, 0, 6, 0, 3, 0, 0, 3, 6, 1, 2, 0, 7, 4, 6, 0, 5, 0, 8, 2, 8, 0, 4, 0, 2, 5, 10, 0, 1, 2, 9, 6, 9, 0, 5, 0, 10, 4, 0, 3, 7, 0, 12, 7, 4, 0, 3, 0, 11, 1, 14, 1, 9, 0, 8, 0, 12, 0, 8, 4, 13, 8, 12, 0, 4, 2, 16, 9, 14, 5, 5, 0, 3, 6, 4

Offset: 1

Author

Gus Wiseman, Jul 26 2019

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), with diagram:
  o o o o o o
  o o o o o .
  o o o o o .
  o o o . . .
The size of the complement (shown in dots) in a 6 X 4 rectangle is 5, so a(7865) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A056239, A061395, A106529, A112798, A268192.

Cf. A316413, A326836, A326837, A326845, A326846, A326848.

Programs

Mathematica

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

PARI

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
A326844(n) = ((bigomega(n)*A061395(n)) - A056239(n)); \\ Antti Karttunen, Feb 10 2023

Formula

a(n) = A001222(n) * A061395(n) - A056239(n).

Extensions

Data section extended up to term a(100) by Antti Karttunen, Feb 10 2023