OFFSET
1,192
COMMENTS
I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..100000
EXAMPLE
The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
3*4*4*4 2*2*2*6*6*6 2*2*4*6*6*6 4*4*4*4*6*6
2*2*2*2*2*6 2*2*3*4*6*6 2*3*4*4*6*6 2*2*2*2*2*6*6*8
2*3*3*4*4*6 3*3*4*4*4*6 2*2*2*2*3*3*8*8
2*2*2*2*3*3*3*8 2*2*2*2*3*4*6*8
2*2*2*2*2*2*2*3*9 2*2*2*3*3*4*4*8
2*2*2*2*2*2*2*8*9
2*2*2*2*2*2*4*4*9
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], Length[#]==Max[#]&]], {n, 100}]
PROG
(PARI) A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ Antti Karttunen, Jun 19 2024
CROSSREFS
The co-balanced version is A340596.
Positions of nonzero terms are A340597.
The case of powers of two is A340611.
Taking maximum Omega instead of maximum factor gives A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 20 2021
EXTENSIONS
Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024
STATUS
approved