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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 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

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