A305052 z-density of the integer partition with Heinz number n. Clutter density of the n-th multiset multisystem (A302242).

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

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

Table of n, a(n) for n=1..66.

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

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

Sequence in context: A328048 A363522 A091090 * A305079 A319841 A336099

Adjacent sequences: A305049 A305050 A305051 * A305053 A305054 A305055

Keyword

sign

Author

Gus Wiseman, May 24 2018

Status

approved