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A106615
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a(n) = numerator of n/(n+14).
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6
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0, 1, 1, 3, 2, 5, 3, 1, 4, 9, 5, 11, 6, 13, 1, 15, 8, 17, 9, 19, 10, 3, 11, 23, 12, 25, 13, 27, 2, 29, 15, 31, 16, 33, 17, 5, 18, 37, 19, 39, 20, 41, 3, 43, 22, 45, 23, 47, 24, 7, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 9, 32, 65, 33, 67, 34, 69, 5, 71, 36, 73, 37, 75, 38, 11, 39
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OFFSET
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0,4
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COMMENTS
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A multiplicative function and also a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. It follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 22 2019
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).
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FORMULA
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Dirichlet g.f.: zeta(s-1)*(1 - 6/7^s - 1/2^s + 6/14^s). - R. J. Mathar, Apr 18 2011
a(n) = 2*a(n-14) - a(n-28). - G. C. Greubel, Feb 19 2019
a(n) = n/gcd(n,14).
a(n) = n/b(n), where b(n) = [1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 14, ...] is a purely periodic sequence of period 14. Thus a(n) is a quasi-polynomial in n.
If gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
O.g.f.: Sum_{d divides 14} A023900(d)*x^d/(1 - x^d)^2 = x/(1 - x)^2 - x^2/(1 - x^2)^2 - 6*x^7/(1 - x^7)^2 + 6*x^14/(1 - x^14)^2.
O.g.f. for reciprocals: Sum_{n >= 1} (1/a(n)*x^n = L(x) + 1/2*L(x^2) + 6/7*L(x^7) + 6/14*L(x^14), where L(x) = log (1/(1 - x)). (End)
Multiplicative with a(2^e) = 2^max(0,e-1), a(7^e) = 7^max(0,e-1), and a(p^e) = p^e otherwise.
Sum_{k=1..n} a(k) ~ (129/392) * n^2. (End)
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MAPLE
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MATHEMATICA
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PROG
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(Sage) [lcm(n, 14)/14 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009
(PARI) vector(100, n, n--; numerator(n/(n+14))) \\ G. C. Greubel, Feb 19 2019
(GAP) List([0..80], n->NumeratorRat(n/(n+14))); # 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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