A257541 - OEIS

%I #28 Feb 14 2020 16:36:35

%S 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,

%T -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,

%U -1,17,9,1,-5,4,2,18,4,7,1,19,-3,20,10,0,5

%N The rank of the partition with Heinz number n.

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

%C 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

%C This is the Dyson rank (St000145), which is different from the Frobenius rank (St000183); see the FindStat links. - _Gus Wiseman_, Apr 13 2019

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

%H Alois P. Heinz, <a href="/A257541/b257541.txt">Table of n, a(n) for n = 2..10000</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000145">St000145: The Dyson rank of a partition</a>

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000183">St000183: The side length of the Durfee square of an integer partition</a>

%F 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.

%e 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.

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

%t Table[PrimePi@ FactorInteger[n][[-1, 1]] - PrimeOmega@ n, {n, 2, 76}] (* _Michael De Vlieger_, May 09 2015 *)

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

%Y Cf. A000720, A001222, A061395, A215366.

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

%K sign

%O 2,4

%A _Emeric Deutsch_, May 09 2015