A257541 The rank of the partition with Heinz number n.

0, 1, -1, 2, 0, 3, -2, 0, 1, 4, -1, 5, 2, 1, -3, 6, -1, 7, 0, 2, 3, 8, -2, 1, 4, -1, 1, 9, 0, 10, -4, 3, 5, 2, -2, 11, 6, 4, -1, 12, 1, 13, 2, 0, 7, 14, -3, 2, 0, 5, 3, 15, -2, 3, 0, 6, 8, 16, -1, 17, 9, 1, -5, 4, 2, 18, 4, 7, 1, 19, -3, 20, 10, 0, 5

Offset: 2

Author

Emeric Deutsch, May 09 2015

Keywords

• sign

Comments

The rank of a partition p is the largest part of p minus the number of parts of p.

The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1,1,1] the Heinz number is 2*2*2 = 8. Its rank is 1 - 3 = -2 = a(8). - Emeric Deutsch, Jun 09 2015

This is the Dyson rank (St000145), which is different from the Frobenius rank (St000183); see the FindStat links. - Gus Wiseman, Apr 13 2019

Examples

			a(24) = -2. Indeed, the partition corresponding to the Heinz number 24 = 2*2*2*3 is [1,1,1,2]; consequently, a(24)= 2 - 4 = -2.
		

References

• G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000
• FindStat, St000145: The Dyson rank of a partition
• FindStat, St000183: The side length of the Durfee square of an integer partition

Crossrefs

Positions of 0's are A106529. Positions of 1's are A325233. Positions of -1's are A325234.

Cf. A000720, A001222, A061395, A215366.

Cf. A046660, A047993, A056239, A093641, A101198, A112798, A174090, A243055, A257990, A263297, A325134, A325178, A325225, A325235.

Programs

Maple

with(numtheory): a := proc(n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: seq(a(n), n = 2 .. 120);

Mathematica

Table[PrimePi@ FactorInteger[n][[-1, 1]] - PrimeOmega@ n, {n, 2, 76}] (* Michael De Vlieger, May 09 2015 *)

Formula

a(n) = q(largest prime factor of n) - bigomega(n), where q(p) is defined by q-th prime = p while bigomega(n) is the number of prime factors of n, including multiplicities.