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A359910 Number of odd-length integer factorizations of n into factors > 1 with the same mean as median.

%I A359910 #13 Jan 20 2025 13:09:35
%S A359910 0,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T A359910 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,
%U A359910 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3
%N A359910 Number of odd-length integer factorizations of n into factors > 1 with the same mean as median.
%C A359910 The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
%H A359910 Antti Karttunen, <a href="/A359910/b359910.txt">Table of n, a(n) for n = 1..65537</a>
%e A359910 The a(n) factorizations for n = 120, 960, 5760, 6720:
%e A359910   120      960         5760            6720
%e A359910   4*5*6    2*16*30     16*18*20        4*30*56
%e A359910   2*6*10   4*12*20     3*5*6*8*8       10*21*32
%e A359910            8*10*12     4*4*6*6*10      12*20*28
%e A359910            3*4*4*4*5   2*2*8*10*18     4*5*6*7*8
%e A359910                        2*2*2*4*4*5*9   2*4*7*10*12
%e A359910                                        2*2*2*4*5*6*7
%t A359910 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A359910 Table[Length[Select[facs[n],OddQ[Length[#]]&&Mean[#]==Median[#]&]],{n,100}]
%o A359910 (PARI) A359910(n, m=n, facs=List([])) = if(1==n, (((#facs)%2) && (facs[(1+#facs)/2]==(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359910(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025
%Y A359910 The version for partitions is A359895, ranked by A359891.
%Y A359910 This is the odd-length case of A359909, partitions A240219.
%Y A359910 A001055 counts factorizations.
%Y A359910 A326622 counts factorizations with integer mean, strict A328966.
%Y A359910 Cf. A316313, A326567/A326568, A359889, A359894, A359897, A359902, A359906, A359911, A360005.
%K A359910 nonn
%O A359910 1,8
%A A359910 _Gus Wiseman_, Jan 24 2023
%E A359910 More terms from _Antti Karttunen_, Jan 20 2025