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A303386 Number of aperiodic factorizations of n > 1.

%I A303386 #9 Sep 25 2018 20:51:28
%S A303386 1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,7,1,2,2,4,1,5,1,6,2,2,2,
%T A303386 7,1,2,2,7,1,5,1,4,4,2,1,12,1,4,2,4,1,7,2,7,2,2,1,11,1,2,4,7,2,5,1,4,
%U A303386 2,5,1,16,1,2,4,4,2,5,1,12,3,2,1,11,2,2,2,7,1,11,2,4,2,2,2,19,1,4,4,7,1,5,1,7,5
%N A303386 Number of aperiodic factorizations of n > 1.
%C A303386 An aperiodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities are relatively prime.
%H A303386 Antti Karttunen, <a href="/A303386/b303386.txt">Table of n, a(n) for n = 2..65537</a>
%F A303386 a(n) = Sum_{d|A052409(n)} mu(d) * A001055(n^(1/d)), where mu = A008683.
%e A303386 The a(36) = 7 aperiodic factorizations are (2*2*9), (2*3*6), (2*18), (3*3*4), (3*12), (4*9), and (36). Missing from this list are (2*2*3*3) and (6*6).
%t A303386 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A303386 Table[Length[Select[facs[n],GCD@@Length/@Split[#]===1&]],{n,2,100}]
%o A303386 (PARI)
%o A303386 A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
%o A303386 A052409(n) = { my(k=ispower(n)); if(k, k, n>1); }; \\ From A052409
%o A303386 A303386(n) = if(1==n,n,my(r); sumdiv(A052409(n),d, ispower(n,d,&r); moebius(d)*A001055(r))); \\ _Antti Karttunen_, Sep 25 2018
%Y A303386 Cf. A000740, a(2^n) = A000837(n), A001055, A007716, A007916, A045778, A052409, A052410, A100953, A162247, A275024, A275870, A281113, A281116, A301700, A302242, A303431.
%K A303386 nonn
%O A303386 2,5
%A A303386 _Gus Wiseman_, Apr 23 2018
%E A303386 More terms from _Antti Karttunen_, Sep 25 2018