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A340599
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Number of factorizations of n into factors > 1 with length and greatest factor equal.
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24
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0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
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OFFSET
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1,192
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COMMENTS
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I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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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.
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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024
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STATUS
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approved
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