%I #12 Feb 02 2019 14:44:08

%S 0,-1,-1,-2,-1,-2,-1,-3,-1,-2,-1,-3,-1,-2,-2,-4,-1,-2,-1,-3,-1,-2,-1,

%T -4,-1,-2,-1,-3,-1,-3,-1,-5,-2,-2,-2,-3,-1,-2,-1,-4,-1,-2,-1,-3,-2,-2,

%U -1,-5,-1,-2,-2,-3,-1,-2,-2,-4,-1,-2,-1,-4,-1,-2,-1,-6,-1,-3

%N z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

%C First nonnegative entry after a(1) = 0 is a(169) = 0.

%e The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.

%e The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.

%e The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.

%e The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.

%t zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];

%t Array[zens,100]

%Y Cf. A001221, A030019, A048143, A056239, A112798, A285572, A286518, A286520, A290103, A302242, A303837, A304118, A304714, A304716, A304911, A305001.

%K sign

%O 1,4

%A _Gus Wiseman_, May 24 2018