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A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

%I A359911 #10 Jan 20 2025 10:40:53
%S A359911 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,2,0,0,0,1,0,1,0,3,0,0,
%T A359911 0,3,0,0,0,3,0,1,0,1,1,0,0,6,0,1,0,1,0,3,0,3,0,0,0,4,0,0,1,4,0,1,0,1,
%U A359911 0,1,0,9,0,0,1,1,0,1,0,6,1,0,0,5,0,0,0,3,0,5,0,1,0,0,0,13,0,1,1,3,0,1,0,3,0,0,0,10
%N A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.
%H A359911 Antti Karttunen, <a href="/A359911/b359911.txt">Table of n, a(n) for n = 1..65537</a>
%F A359911 For n > 1, a(n) = A001055(n) - A359909(n). - _Antti Karttunen_, Jan 20 2025
%e A359911 The a(72) = 9 factorizations: (2*2*2*3*3), (2*2*2*9), (2*2*3*6), (2*2*18), (2*3*12), (2*4*9), (2*6*6), (3*3*8), (3*4*6).
%t A359911 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A359911 Table[Length[Select[facs[n],Mean[#]!=Median[#]&]],{n,100}]
%o A359911 (PARI)
%o A359911 median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
%o A359911 A359911(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)!=(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359911(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A359911 The version for partitions is A359894, complement A240219.
%Y A359911 The complement is counted by A359909, odd-length A359910.
%Y A359911 A001055 counts factorizations.
%Y A359911 A326622 counts factorizations with integer mean, strict A328966.
%Y A359911 Cf. A316313, A326567/A326568, A359897, A359889, A359906, A360005.
%K A359911 nonn
%O A359911 1,24
%A A359911 _Gus Wiseman_, Jan 24 2023
%E A359911 Data section extended to a(108) by _Antti Karttunen_, Jan 20 2025