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Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).
202

%I #11 Nov 10 2021 17:27:08

%S 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 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 3,3,3,2,2,3,3,3,3,3,2,1,4,1,1,3,2,2,3,1,4

%N Total weight of the n-th multiset multisystem. Totally additive with a(prime(n)) = Omega(n).

%C 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 Alois P. Heinz, <a href="/A302242/b302242.txt">Table of n, a(n) for n = 1..65536</a>

%e 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 with(numtheory):

%p a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Sep 07 2018

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Total[PrimeOmega/@primeMS[n]],{n,100}]

%o (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 Cf. A001222, A003963, A007716, A034691, A056239, A061775, A063834, A096443, A249620, A255397, A255906, A275024, A279789, A281113, A299757, A302243.

%K nonn

%O 1,7

%A _Gus Wiseman_, Apr 03 2018