A359909 - OEIS

%I #13 Jan 20 2025 09:07:46

%S 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2,2,1,5,2,2,3,3,1,4,1,4,2,2,

%T 2,6,1,2,2,4,1,4,1,3,3,2,1,6,2,3,2,3,1,4,2,4,2,2,1,7,1,2,3,7,2,4,1,3,

%U 2,4,1,7,1,2,3,3,2,4,1,6,4,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,3,3,6,1,4,1,4,5,2,1,6,1,4,2,5,1,4,2,3,3,2,2,11

%N Number of integer factorizations of n into factors > 1 with the same mean as median.

%C 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 Antti Karttunen, <a href="/A359909/b359909.txt">Table of n, a(n) for n = 1..65537</a>

%e The a(n) factorizations for n = 24, 36, 60, 120, 144, 360:

%e   24      36        60      120       144       360

%e   3*8     4*9       2*30    2*60      2*72      4*90

%e   4*6     6*6       3*20    3*40      3*48      5*72

%e   2*12    2*18      4*15    4*30      4*36      6*60

%e   2*3*4   3*12      5*12    5*24      6*24      8*45

%e           2*2*3*3   6*10    6*20      8*18      9*40

%e                     3*4*5   8*15      9*16      10*36

%e                             10*12     12*12     12*30

%e                             4*5*6     2*2*6*6   15*24

%e                             2*6*10    3*3*4*4   18*20

%e                             2*3*4*5             2*180

%e                                                 3*120

%e                                                 2*10*18

%e                                                 3*4*5*6

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Select[facs[n],Mean[#]==Median[#]&]],{n,100}]

%o (PARI)

%o median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);

%o A359909(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 += A359909(n/d, d, newfacs))); (s)); \\ _Antti Karttunen_, Jan 20 2025

%Y The version for partitions is A240219, complement A359894.

%Y These multisets are ranked by A359889.

%Y The version for strict partitions is A359897.

%Y The odd-length case is A359910.

%Y The complement is counted by A359911.

%Y A001055 counts factorizations.

%Y A058398 counts partitions by mean, see also A008284, A327482.

%Y A326622 counts factorizations with integer mean, strict A328966.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y Cf. A316313, A326567/A326568, A359906, A360005.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jan 24 2023

%E More terms from _Antti Karttunen_, Jan 20 2025