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A357877
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The a(n)-th composition in standard order is the sequence of run-sums of the prime indices of n.
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0
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0, 1, 2, 2, 4, 6, 8, 4, 8, 12, 16, 10, 32, 24, 20, 8, 64, 24, 128, 20, 40, 48, 256, 18, 32, 96, 32, 40, 512, 52, 1024, 16, 80, 192, 72, 40, 2048, 384, 160, 36, 4096, 104, 8192, 80, 68, 768, 16384, 34, 128, 96, 320, 160, 32768, 96, 144, 72, 640, 1536, 65536, 84
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OFFSET
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1,3
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COMMENTS
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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.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).
The k-th composition in standard order (graded reverse-lexicographic, 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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EXAMPLE
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The prime indices of 24 are (1,1,1,2), with run-sums (3,2), and this is the 18th composition in standard order, so a(24) = 18.
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[primeMS[n]]], {n, 100}]
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CROSSREFS
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The version for prime indices instead of standard compositions is A353832.
The version for standard compositions instead of prime indices is A353847.
A066099 lists standard compositions.
A351014 counts distinct runs in standard compositions.
Cf. A118914, A181819, A238279, A239312, A275870, A300273, A304405, A304442, A304660, A333755, A353743-A354912, A357875.
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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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