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.

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

Author

Tian Vlasic, Jan 29 2024

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A001055, A002033, A025487, A045778, A050322, A064554, A077565, A097296, A190938, A216599, A317146.