OFFSET
1,12
COMMENTS
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
EXAMPLE
The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
2*6 4*6 4*9 2*4*9 4*4*9 8*30
3*4 2*2*6 6*6 2*6*6 4*6*6 12*20
2*3*4 2*2*9 3*4*6 2*2*4*9 5*6*8
2*3*6 2*2*2*9 2*2*6*6 2*4*30
3*3*4 2*2*3*6 2*3*4*6 2*6*20
2*3*3*4 3*3*4*4 2*8*15
2*2*2*2*9 3*4*20
2*2*2*3*6 3*8*10
2*2*3*3*4 4*5*12
2*10*12
2*3*5*8
2*2*2*30
2*2*3*20
2*2*5*12
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], #=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]], {n, 100}]
PROG
(PARI) A340654(n, m=n, om=omega(n), mbo=0) = if(1==n, (mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo, bigomega(d)))))); \\ Antti Karttunen, Jun 19 2024
CROSSREFS
Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340651.
The balanced version is A340653.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 15 2021
EXTENSIONS
Data section extended up to a(105) by Antti Karttunen, Jun 19 2024
STATUS
approved