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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.

%I A369713 #28 Feb 18 2024 12:34:03
%S A369713 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,
%T A369713 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,
%U A369713 1,4,2,6,1,24,1,2,4,4,2,6,1,16,5,2,1,16,2
%N 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.
%C A369713 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.
%C A369713 For every natural number n, a(n) only depends on the prime signature of n.
%C A369713 a(n) is even if and only if n is a composite number.
%C A369713 Conjecture: There exists c such that a(n) <= n^c for all natural numbers n.
%e A369713 The factorizations of 60 followed by their Moebius values are the following:
%e A369713  (2*2*3*5) -> -3
%e A369713  (2*2*15) ->  1
%e A369713  (2*3*10) ->  2
%e A369713  (2*5*6) ->  2
%e A369713  (2*30) -> -1
%e A369713  (3*4*5) ->  2
%e A369713  (3*20) -> -1
%e A369713  (4*15) -> -1
%e A369713  (5*12) -> -1
%e A369713  (6*10) -> -1
%e A369713  (60) ->  1
%e A369713 Thus a(60)=16.
%Y A369713 Cf. A001055, A002033, A025487, A045778, A050322, A064554, A077565, A097296, A190938, A216599, A317146.
%K A369713 nonn
%O A369713 1,4
%A A369713 _Tian Vlasic_, Jan 29 2024