OFFSET
2,4
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
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
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.
EXAMPLE
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.
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 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Emeric Deutsch, May 09 2015
STATUS
approved