OFFSET
1,4
COMMENTS
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.
LINKS
Mathematics Stack Exchange, Why does mathematical convention deal so ineptly with multisets?
Wikiversity, Partitions of multisets
FORMULA
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.
EXAMPLE
The sequence of multiset partitions begins:
(), ((1)), ((2)), ((1)(1)), ((11)), ((1)(2)), ((3)),
((1)(1)(1)), ((2)(2)), ((1)(11)), ((12)), ((1)(1)(2)),
((4)), ((1)(3)), ((2)(11)), ((1)(1)(1)(1)), ((111)),
((1)(2)(2)), ((22)), ((1)(1)(11)), ((2)(3)), ((1)(12)),
((13)), ((1)(1)(1)(2)), ((11)(11)), ((1)(4)), ((2)(2)(2)),
((1)(1)(3)), ((5)), ((1)(2)(11)), ((112)), ((1)(1)(1)(1)(1)),
((2)(12)), ((1)(111)), ((3)(11)), ((1)(1)(2)(2)), ((6)), ...
MATHEMATICA
Table[Total[Cases[FactorInteger[n], {p_, k_}:>k*PrimeOmega[PrimePi[p]+1]]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 12 2016
STATUS
approved