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A106608
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a(n) = numerator of n/(n+7).
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21
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0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76
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OFFSET
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0,3
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COMMENTS
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In general, the numerators of n/(n+p) for prime p and n >= 0 form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - Paul D. Hanna, Jul 27 2005
Generalizing the above comment of Hanna, the numerators of n/(n + k) for a fixed positive integer k and n >= 0 form a sequence with the g.f.: Sum_{d divides k} f(d)*x^d/(1 - x^d)^2, where f(n) is the Dirichlet inverse of the Euler totient function A000010. f(n) is a multiplicative function defined on prime powers p^k by f(p^k) = 1 - p. See A023900. - Peter Bala, Feb 17 2019
a(n) <> n iff n = 7 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).
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FORMULA
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Dirichlet g.f.: zeta(s-1)*(1-6/7^s).
Multiplicative with a(p^e) = p^e if p<>7, a(7^e) = 7^(e-1) if e > 0. (End)
Sum_{k=1..n} a(k) ~ (43/98) * n^2. - Amiram Eldar, Nov 25 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 13*log(2)/7. - Amiram Eldar, Sep 08 2023
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MAPLE
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MATHEMATICA
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PROG
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(PARI) vector(100, n, n--; numerator(n/(n+7))) \\ G. C. Greubel, Feb 19 2019
(GAP) List([0..80], n->NumeratorRat(n/(n+7))); # Muniru A Asiru, Feb 19 2019
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CROSSREFS
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KEYWORD
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nonn,easy,frac,mult
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AUTHOR
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STATUS
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approved
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