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A348611
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Number of ordered factorizations of n with no adjacent equal factors.
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8
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1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 14, 1, 3, 3, 6, 1, 13, 1, 7, 3, 3, 3, 17, 1, 3, 3, 14, 1, 13, 1, 6, 6, 3, 1, 29, 1, 6, 3, 6, 1, 14, 3, 14, 3, 3, 1, 36, 1, 3, 6, 14, 3, 13, 1, 6, 3, 13, 1, 45, 1, 3, 6, 6, 3, 13, 1, 29, 4, 3
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OFFSET
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1,6
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COMMENTS
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An ordered factorization of n is a finite sequence of positive integers > 1 with product n.
In analogy with Carlitz compositions, these may be called Carlitz ordered factorizations.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) ordered factorizations without adjacent equal factors for n = 1, 6, 12, 16, 24, 30, 32, 36 are:
() 6 12 16 24 30 32 36
2*3 2*6 2*8 3*8 5*6 4*8 4*9
3*2 3*4 8*2 4*6 6*5 8*4 9*4
4*3 2*4*2 6*4 10*3 16*2 12*3
6*2 8*3 15*2 2*16 18*2
2*3*2 12*2 2*15 2*8*2 2*18
2*12 3*10 4*2*4 3*12
2*3*4 2*3*5 2*3*6
2*4*3 2*5*3 2*6*3
2*6*2 3*2*5 2*9*2
3*2*4 3*5*2 3*2*6
3*4*2 5*2*3 3*4*3
4*2*3 5*3*2 3*6*2
4*3*2 6*2*3
6*3*2
2*3*2*3
3*2*3*2
Thus, of total A074206(12) = 8 ordered factorizations of 12, only factorizations 2*2*3 and 3*2*2 (see A348616) are not included in this count, therefore a(12) = 6. - Antti Karttunen, Nov 12 2021
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MATHEMATICA
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ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[Prepend[#, d]&/@ordfacs[n/d], {d, Rest[Divisors[n]]}]];
antirunQ[y_]:=Length[y]==Length[Split[y]]
Table[Length[Select[ordfacs[n], antirunQ]], {n, 100}]
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PROG
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CROSSREFS
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Factorizations without a permutation of this type are counted by A333487.
Factorizations with a permutation of this type are counted by A335434.
The alternating case is A348610, which is dominated at positions A122181.
The complement is counted by A348616.
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A348613 counts non-alternating ordered factorizations.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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