A275024 Total weight of the n-th twice-prime-factored multiset partition.

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

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

Table of n, a(n) for n=1..104.

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