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A369713
a(n) is the sum over all multiplicative partitions k of n of the absolute value of the Möbius function evaluated at k,n in the poset of multiplicative partitions of n under refinement.
0
1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 2, 4, 1, 6, 1, 6, 2, 2, 2, 11, 1, 2, 2, 8, 1, 6, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 16, 1, 2, 4, 11, 2, 6, 1, 4, 2, 6, 1, 24, 1, 2, 4, 4, 2, 6, 1, 16, 5, 2, 1, 16, 2
OFFSET
1,4
COMMENTS
If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.
For every natural number n, a(n) only depends on the prime signature of n.
a(n) is even if and only if n is a composite number.
Conjecture: There exists c such that a(n) <= n^c for all natural numbers n.
EXAMPLE
The factorizations of 60 followed by their Moebius values are the following:
(2*2*3*5) -> -3
(2*2*15) -> 1
(2*3*10) -> 2
(2*5*6) -> 2
(2*30) -> -1
(3*4*5) -> 2
(3*20) -> -1
(4*15) -> -1
(5*12) -> -1
(6*10) -> -1
(60) -> 1
Thus a(60)=16.
KEYWORD
nonn
AUTHOR
Tian Vlasic, Jan 29 2024
STATUS
approved