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 A275024 Total weight of the n-th twice-prime-factored multiset partition. 53
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A007716, A034691, A096443, A255906, A249620. Sequence in context: A091222 A316506 A294884 * A325121 A064659 A349520 Adjacent sequences: A275021 A275022 A275023 * A275025 A275026 A275027 KEYWORD nonn AUTHOR Gus Wiseman, Nov 12 2016 STATUS approved

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Last modified February 8 20:00 EST 2023. Contains 360152 sequences. (Running on oeis4.)