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A334031
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The smallest number whose unsorted prime signature is the reversed n-th composition in standard order.
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3
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1, 2, 4, 6, 8, 18, 12, 30, 16, 54, 36, 150, 24, 90, 60, 210, 32, 162, 108, 750, 72, 450, 300, 1470, 48, 270, 180, 1050, 120, 630, 420, 2310, 64, 486, 324, 3750, 216, 2250, 1500, 10290, 144, 1350, 900, 7350, 600, 4410, 2940, 25410, 96, 810, 540, 5250, 360, 3150
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OFFSET
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0,2
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COMMENTS
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All terms are normal (A055932), meaning their prime indices cover an initial interval of positive integers.
Unsorted prime signature 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 sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
18: {1,2,2}
12: {1,1,2}
30: {1,2,3}
16: {1,1,1,1}
54: {1,2,2,2}
36: {1,1,2,2}
150: {1,2,3,3}
24: {1,1,1,2}
90: {1,2,2,3}
60: {1,1,2,3}
210: {1,2,3,4}
32: {1,1,1,1,1}
162: {1,2,2,2,2}
For example, the 13th composition in standard order is (1,2,1), and the least number with prime signature (1,2,1) is 90 = 2^1 * 3^2 * 5^1, so a(13) = 90.
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Product[Prime[i]^stc[n][[-i]], {i, DigitCount[n, 2, 1]}], {n, 0, 100}]
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CROSSREFS
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The non-reversed version is A057335.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Normal numbers with standard compositions as prime signature are A334032.
All of the following pertain to compositions in standard order (A066099):
- Constant compositions are A272919.
- Aperiodic compositions are A328594.
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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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