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A340654
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Number of cross-balanced factorizations of n.
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23
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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, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 5, 1, 1, 1, 3, 1
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OFFSET
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1,12
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COMMENTS
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We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).
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LINKS
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EXAMPLE
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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
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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], #=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]], {n, 100}]
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PROG
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(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
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CROSSREFS
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Positions of terms > 1 are A126706.
The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340651.
The twice-balanced version is A340655.
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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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