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A320089
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Number of primitive (=aperiodic) 5-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.
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3
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1, 5, 29, 149, 773, 3869, 19493, 97493, 488093, 2440589, 12206213, 61031093, 305171717, 1525859213, 7629374189, 38146874189, 190734764813, 953673824213, 4768371089837, 23841855464717, 119209287089693, 596046435527189, 2980232226542813, 14901161132714813
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = Sum_{j=1..n} Sum_{d|j} 5^(d-1) * mu(j/d).
G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 5*x^k). - Ilya Gutkovskiy, Dec 11 2020
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MAPLE
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b:= n-> add(`if`(d=n, 5^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);
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MATHEMATICA
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nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)
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PROG
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(PARI) a(n) = sum(j=1, n, sumdiv(j, d, 5^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020
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CROSSREFS
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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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