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A334033
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The a(n)-th composition in standard order (graded reverse-lexicographic) is the reversed unsorted prime signature of n.
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3
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0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 6, 1, 3, 3, 8, 1, 5, 1, 6, 3, 3, 1, 12, 2, 3, 4, 6, 1, 7, 1, 16, 3, 3, 3, 10, 1, 3, 3, 12, 1, 7, 1, 6, 6, 3, 1, 24, 2, 5, 3, 6, 1, 9, 3, 12, 3, 3, 1, 14, 1, 3, 6, 32, 3, 7, 1, 6, 3, 7, 1, 20, 1, 3, 5, 6, 3, 7, 1, 24, 8, 3, 1
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OFFSET
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1,4
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COMMENTS
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Unsorted prime signature (A124010) is the sequence of exponents in a number's prime factorization.
The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
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LINKS
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FORMULA
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EXAMPLE
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The unsorted prime signature of 12345678 is (1,2,1,1), whose reverse (1,1,2,1) is the 29th composition in standard order, so a(12345678) = 29.
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MATHEMATICA
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stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
Table[stcinv[Reverse[Last/@If[n==1, {}, FactorInteger[n]]]], {n, 100}]
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CROSSREFS
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Positions of first appearances are A334031.
The non-reversed version is A334032.
Unsorted prime signature is A124010.
Least number with reversed prime signature is A331580.
All of the following pertain to compositions in standard order (A066099):
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
Cf. A029931, A048793, A052409, A055932, A056239, A057335, A071364, A112798, A124767, A228351, A233249, A329139, A333220.
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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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