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

1,4

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.

Links

Table of n, a(n) for n=1..85.

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.

Crossrefs

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

Sequence in context: A370814 A347050 A348383 * A295635 A115751 A324888

Adjacent sequences: A369710 A369711 A369712 * A369714 A369715 A369716

Keyword

nonn

Author

Tian Vlasic, Jan 29 2024

Status

approved