login
Number of balanced factorizations of n.
42

%I #14 Oct 22 2023 15:13:38

%S 1,1,1,0,1,0,1,1,0,0,1,2,1,0,0,1,1,2,1,2,0,0,1,1,0,0,1,2,1,3,1,1,0,0,

%T 0,2,1,0,0,1,1,3,1,2,2,0,1,2,0,2,0,2,1,1,0,1,0,0,1,2,1,0,2,1,0,3,1,2,

%U 0,3,1,3,1,0,2,2,0,3,1,2,1,0,1,2,0,0,0,1,1,2,0,2,0,0,0,3,1,2,2,2,1,3,1,1,3,0,1,3,1,3,0,2,1,3,0,2,2,0,0,4

%N Number of balanced factorizations of n.

%C A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

%H Antti Karttunen, <a href="/A340653/b340653.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%e The balanced factorizations for n = 120, 144, 192, 288, 432, 768:

%e 3*5*8 2*8*9 3*8*8 4*8*9 6*8*9 8*8*12

%e 2*2*30 3*6*8 4*6*8 6*6*8 2*8*27 2*2*8*24

%e 2*3*20 2*4*18 2*8*12 2*8*18 3*8*18 2*3*8*16

%e 2*5*12 2*6*12 4*4*12 3*8*12 4*4*27 2*4*4*24

%e 3*4*12 2*2*2*24 4*4*18 4*6*18 2*4*6*16

%e 2*2*3*16 4*6*12 4*9*12 3*4*4*16

%e 2*12*12 6*6*12 2*2*12*16

%e 2*2*2*36 2*12*18 2*2*2*2*48

%e 2*2*3*24 3*12*12 2*2*2*3*32

%e 2*3*3*16 2*2*2*54

%e 2*2*3*36

%e 2*3*3*24

%e 3*3*3*16

%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],#=={}||Length[#]==Max[PrimeOmega/@#]&]],{n,100}]

%o (PARI) A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ _Antti Karttunen_, Oct 22 2023

%Y Positions of zeros are A001358.

%Y Positions of nonzero terms are A100959.

%Y The co-balanced version is A340596.

%Y Taking maximum factor instead of maximum Omega gives A340599.

%Y The cross-balanced version is A340654.

%Y The twice-balanced version is A340655.

%Y A001055 counts factorizations.

%Y A045778 counts strict factorizations.

%Y A316439 counts factorizations by product and length.

%Y A320655 counts factorizations into semiprimes.

%Y Other balance-related sequences:

%Y - A010054 counts balanced strict partitions.

%Y - A047993 counts balanced partitions.

%Y - A098124 counts balanced compositions.

%Y - A106529 lists Heinz numbers of balanced partitions.

%Y - A340597 have an alt-balanced factorization.

%Y - A340598 counts balanced set partitions.

%Y - A340600 counts unlabeled balanced multiset partitions.

%Y - A340656 have no twice-balanced factorizations.

%Y - A340657 have a twice-balanced factorization.

%Y Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.

%K nonn

%O 1,12

%A _Gus Wiseman_, Jan 15 2021

%E Data section extended up to a(120) by _Antti Karttunen_, Oct 22 2023