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A348616
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Number of ordered factorizations of n with adjacent equal factors.
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4
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0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 6, 1, 0, 1, 2, 0, 0, 0, 9, 0, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 19, 1, 2, 0, 2, 0, 6, 0, 6, 0, 0, 0, 8, 0, 0, 2, 18, 0, 0, 0, 2, 0, 0, 0, 31, 0, 0, 2, 2, 0, 0, 0, 19, 4, 0, 0, 8, 0, 0
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OFFSET
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1,12
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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.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) ordered factorizations with at least one pair of adjacent equal factors for n = 12, 24, 36, 60:
2*2*3 2*2*6 6*6 15*2*2
3*2*2 6*2*2 2*2*9 2*2*15
2*2*2*3 3*3*4 2*2*3*5
2*2*3*2 4*3*3 2*2*5*3
2*3*2*2 9*2*2 3*2*2*5
3*2*2*2 2*2*3*3 3*5*2*2
2*3*3*2 5*2*2*3
3*2*2*3 5*3*2*2
3*3*2*2
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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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CROSSREFS
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Positions of 4's appear to be A030514.
Positions of 2's appear to be A054753.
Positions of 1's appear to be A168363.
Factorizations with a permutation of this type are counted by A333487.
Factorizations without a permutation of this type are counted by A335434.
The complement is counted by A348611.
Dominated by A348613 (non-alternating ordered factorizations).
A335452 counts anti-run permutations of prime indices, complement A336107.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
Cf. A001250, A025047, A122181, A138364, A347050, A347463, A347706, A348379, A348380, A348382, A348610.
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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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