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A356603
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Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).
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12
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1, 2, 4, 10, 8, 20, 50, 110, 16, 40, 100, 220, 250, 550, 1210, 1870, 32, 80, 200, 440, 500, 1100, 2420, 3740, 1250, 2750, 6050, 9350, 13310, 20570, 31790, 43010, 64, 160, 400, 880, 1000, 2200, 4840, 7480, 2500, 5500, 12100, 18700, 26620, 41140, 63580, 86020
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OFFSET
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0,2
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COMMENTS
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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.
The image consists of all numbers whose prime indices are odd and cover an initial interval of odd positive integers.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
10: {1,3}
8: {1,1,1}
20: {1,1,3}
50: {1,3,3}
110: {1,3,5}
16: {1,1,1,1}
40: {1,1,1,3}
100: {1,1,3,3}
220: {1,1,3,5}
250: {1,3,3,3}
550: {1,3,3,5}
1210: {1,3,5,5}
1870: {1,3,5,7}
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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_]:=1/2 Total[2^Accumulate[Reverse[q]]];
mnrm[s_]:=If[Min@@s==1, mnrm[DeleteCases[s-1, 0]]+1, 0];
sq=stcinv/@Table[Length/@Split[primeMS[n], #1>=#2-1&], {n, 1000}];
Table[Position[sq, k][[1, 1]], {k, 0, mnrm[Rest[sq]]}]
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CROSSREFS
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See link for sequences related to standard compositions.
The partitions with these Heinz numbers are counted by A053251.
A subset of A066208 (numbers with all odd prime indices).
Up to permutation, these are the positions of first appearances of rows in A356226. Other statistics are:
An ordered version is counted by A356604.
Cf. A000005, A001222, A055932, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.
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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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