This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A340596 #20 Jun 10 2024 10:01:50
%S A340596 1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,3,1,1,1,2,1,1,1,1,1,1,
%T A340596 1,4,1,1,1,3,1,1,1,2,2,1,1,4,1,2,1,2,1,3,1,3,1,1,1,4,1,1,2,1,1,1,1,2,
%U A340596 1,1,1,5,1,1,2,2,1,1,1,4,1,1,1,4,1,1,1,3,1,4,1,2,1,1,1,5,1,2,2,4,1,1,1,3,1,1,1,5,1,1,1,4,1,1,1,2,2,1,1,8
%N A340596 Number of co-balanced factorizations of n.
%C A340596 We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.
%H A340596 Antti Karttunen, <a href="/A340596/b340596.txt">Table of n, a(n) for n = 1..65537</a>
%e A340596 The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
%e A340596 2*6 3*8 4*9 8*9 3*5*8 2*72 4*5*9
%e A340596 3*4 4*6 6*6 2*36 4*5*6 3*48 5*6*6
%e A340596 2*12 2*18 3*24 2*2*30 4*36 2*2*45
%e A340596 3*12 4*18 2*3*20 6*24 2*3*30
%e A340596 6*12 2*4*15 8*18 2*5*18
%e A340596 2*5*12 9*16 2*6*15
%e A340596 2*6*10 12*12 2*9*10
%e A340596 3*4*10 3*3*20
%e A340596 3*4*15
%e A340596 3*5*12
%e A340596 3*6*10
%t A340596 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A340596 Table[Length[Select[facs[n],Length[#]==PrimeNu[n]&]],{n,100}]
%o A340596 (PARI) A340596(n, m=n, om=omega(n)) = if(1==n,(0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ _Antti Karttunen_, Jun 10 2024
%Y A340596 Positions of terms > 1 are A126706.
%Y A340596 Positions of 1's are A303554.
%Y A340596 The version for unlabeled multiset partitions is A319616.
%Y A340596 The alt-balanced version is A340599.
%Y A340596 The balanced version is A340653.
%Y A340596 The cross-balanced version is A340654.
%Y A340596 The twice-balanced version is A340655.
%Y A340596 A001055 counts factorizations.
%Y A340596 A045778 counts strict factorizations.
%Y A340596 A316439 counts factorizations by product and length.
%Y A340596 Other balance-related sequences:
%Y A340596 - A010054 counts balanced strict partitions.
%Y A340596 - A047993 counts balanced partitions.
%Y A340596 - A098124 counts balanced compositions.
%Y A340596 - A106529 lists Heinz numbers of balanced partitions.
%Y A340596 - A340597 lists numbers with an alt-balanced factorization.
%Y A340596 - A340598 counts balanced set partitions.
%Y A340596 - A340600 counts unlabeled balanced multiset partitions.
%Y A340596 Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.
%K A340596 nonn
%O A340596 1,12
%A A340596 _Gus Wiseman_, Jan 15 2021
%E A340596 Data section extended up to a(120) by _Antti Karttunen_, Jun 10 2024