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%I A302242 #11 Nov 10 2021 17:27:08
%S A302242 0,0,1,0,1,1,2,0,2,1,1,1,2,2,2,0,1,2,3,1,3,1,2,1,2,2,3,2,2,2,1,0,2,1,
%T A302242 3,2,3,3,3,1,1,3,2,1,3,2,2,1,4,2,2,2,4,3,2,2,4,2,1,2,3,1,4,0,3,2,1,1,
%U A302242 3,3,3,2,2,3,3,3,3,3,2,1,4,1,1,3,2,2,3,1,4
%N A302242 Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).
%C A302242 A multiset multisystem is a finite multiset of finite multisets of positive integers. The n-th multiset multisystem is constructed by factoring n into prime numbers and then factoring each prime index into prime numbers and taking their prime indices. This produces a unique multiset multisystem for each n, and every possible multiset multisystem is so constructed as n ranges over all positive integers.
%H A302242 Alois P. Heinz, <a href="/A302242/b302242.txt">Table of n, a(n) for n = 1..65536</a>
%e A302242 Sequence of finite multisets of finite multisets of positive integers begins: (), (()), ((1)), (()()), ((2)), (()(1)), ((11)), (()()()), ((1)(1)), (()(2)), ((3)), (()()(1)), ((12)), (()(11)), ((1)(2)), (()()()()), ((4)), (()(1)(1)), ((111)), (()()(2)).
%p A302242 with(numtheory):
%p A302242 a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
%p A302242 seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 07 2018
%t A302242 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302242 Table[Total[PrimeOmega/@primeMS[n]],{n,100}]
%o A302242 (PARI) a(n,f=factor(n))=sum(i=1,#f~, bigomega(primepi(f[i,1]))*f[i,2]) \\ _Charles R Greathouse IV_, Nov 10 2021
%Y A302242 Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.
%K A302242 nonn
%O A302242 1,7
%A A302242 _Gus Wiseman_, Apr 03 2018

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