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A333220
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The number k such that the k-th composition in standard order consists of the prime indices of n in weakly increasing order.
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22
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0, 1, 2, 3, 4, 6, 8, 7, 10, 12, 16, 14, 32, 24, 20, 15, 64, 26, 128, 28, 40, 48, 256, 30, 36, 96, 42, 56, 512, 52, 1024, 31, 80, 192, 72, 58, 2048, 384, 160, 60, 4096, 104, 8192, 112, 84, 768, 16384, 62, 136, 100, 320, 224, 32768, 106, 144, 120, 640, 1536
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OFFSET
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1,3
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COMMENTS
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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.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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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 the corresponding compositions begins:
0: () 128: (8) 2048: (12)
1: (1) 28: (1,1,3) 384: (1,8)
2: (2) 40: (2,4) 160: (2,6)
3: (1,1) 48: (1,5) 60: (1,1,1,3)
4: (3) 256: (9) 4096: (13)
6: (1,2) 30: (1,1,1,2) 104: (1,2,4)
8: (4) 36: (3,3) 8192: (14)
7: (1,1,1) 96: (1,6) 112: (1,1,5)
10: (2,2) 42: (2,2,2) 84: (2,2,3)
12: (1,3) 56: (1,1,4) 768: (1,9)
16: (5) 512: (10) 16384: (15)
14: (1,1,2) 52: (1,2,3) 62: (1,1,1,1,2)
32: (6) 1024: (11) 136: (4,4)
24: (1,4) 31: (1,1,1,1,1) 100: (1,3,3)
20: (2,3) 80: (2,5) 320: (2,7)
15: (1,1,1,1) 192: (1,7) 224: (1,1,6)
64: (7) 72: (3,4) 32768: (16)
26: (1,2,2) 58: (1,1,2,2) 106: (1,2,2,2)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
fbi[q_]:=If[q=={}, 0, Total[2^q]/2];
Table[fbi[Accumulate[Reverse[primeMS[n]]]], {n, 100}]
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CROSSREFS
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The version with prime indices taken in weakly decreasing order is A233249.
Cf. A000120, A029931, A048793, A056239, A066099, A070939, A112798, A114994, A225620, A228351, A333221.
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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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