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%I #23 Aug 12 2022 09:17:15
%S 0,1,1,2,2,2,1,3,2,3,2,3,1,2,3,4,3,3,2,4,2,3,2,4,4,2,3,3,1,4,3,5,3,4,
%T 3,4,1,3,2,5,2,3,2,4,4,3,4,5,2,5,4,3,1,4,4,4,3,2,3,5,1,4,3,6,3,4,3,5,
%U 3,4,2,5,2,2,5,4,3,3,1,6,4,3,4,4,5,3,2,5,2,5,2,4,4,5,4,6,2,3,4,6,3,5,3,4
%N Total weight of the n-th twice-prime-factored multiset partition.
%C A multiset partition is a finite multiset of finite nonempty multisets of positive integers. The n-th twice-prime-factored multiset partition is constructed by factoring n into prime numbers and then factoring each prime index plus 1 into prime numbers. This produces a unique multiset of multisets of prime numbers which can then be normalized (see example) to produce each possible multiset partition as n ranges over all positive integers.
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/q/152223">Why does mathematical convention deal so ineptly with multisets?</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Partitions_of_multisets">Partitions of multisets</a>
%F If prime(k) has weight equal to the number of prime factors (counting multiplicity) of k+1, then a(n) is the sum of weights over all prime factors (counting multiplicity) of n.
%e The sequence of multiset partitions begins:
%e (), ((1)), ((2)), ((1)(1)), ((11)), ((1)(2)), ((3)),
%e ((1)(1)(1)), ((2)(2)), ((1)(11)), ((12)), ((1)(1)(2)),
%e ((4)), ((1)(3)), ((2)(11)), ((1)(1)(1)(1)), ((111)),
%e ((1)(2)(2)), ((22)), ((1)(1)(11)), ((2)(3)), ((1)(12)),
%e ((13)), ((1)(1)(1)(2)), ((11)(11)), ((1)(4)), ((2)(2)(2)),
%e ((1)(1)(3)), ((5)), ((1)(2)(11)), ((112)), ((1)(1)(1)(1)(1)),
%e ((2)(12)), ((1)(111)), ((3)(11)), ((1)(1)(2)(2)), ((6)), ...
%t Table[Total[Cases[FactorInteger[n],{p_,k_}:>k*PrimeOmega[PrimePi[p]+1]]],{n,1,100}]
%Y Cf. A007716, A034691, A096443, A255906, A249620.
%K nonn
%O 1,4
%A _Gus Wiseman_, Nov 12 2016
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