OFFSET
1,7
COMMENTS
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..65536
EXAMPLE
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)).
MAPLE
with(numtheory):
a:= n-> add(add(j[2], j=ifactors(pi(i[1]))[2])*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Sep 07 2018
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[PrimeOmega/@primeMS[n]], {n, 100}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 03 2018
STATUS
approved