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A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242). 23
0, -1, -1, -2, -1, -2, -1, -3, -1, -2, -1, -3, -1, -2, -2, -4, -1, -2, -1, -3, -1, -2, -1, -4, -1, -2, -1, -3, -1, -3, -1, -5, -2, -2, -2, -3, -1, -2, -1, -4, -1, -2, -1, -3, -2, -2, -1, -5, -1, -2, -2, -3, -1, -2, -2, -4, -1, -2, -1, -4, -1, -2, -1, -6, -1, -3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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.
First nonnegative entry after a(1) = 0 is a(169) = 0.
LINKS
EXAMPLE
The 1105th multiset multisystem is {{2},{1,2},{4}} with clutter density -2, so a(1105) = -2.
The 5429th multiset multisystem is {{1,2,2},{1,1,1,2}} with clutter density 0, so a(5429) = 0.
The 11837th multiset multisystem is {{1,1},{1,1,1},{1,1,1,2}} with clutter density -1, so a(11837) = -1.
The 42601th multiset multisystem is {{1,2},{1,3},{1,2,3}} with clutter density 1, so a(42601) = 1.
MATHEMATICA
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]]]];
Array[zens, 100]
CROSSREFS
Sequence in context: A328048 A363522 A091090 * A305079 A319841 A336099
KEYWORD
sign
AUTHOR
Gus Wiseman, May 24 2018
STATUS
approved

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Last modified February 26 09:21 EST 2024. Contains 370338 sequences. (Running on oeis4.)