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A106611
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a(n) = numerator of n/(n+10).
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6
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0, 1, 1, 3, 2, 1, 3, 7, 4, 9, 1, 11, 6, 13, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 13, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 37, 19, 39, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 26, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 13, 33, 67, 34, 69, 7, 71, 36, 73, 37, 15, 38, 77, 39
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OFFSET
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0,4
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COMMENTS
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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 17 2019
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).
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FORMULA
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Dirichlet g.f.: zeta(s-1)*(1 - 4/5^s - 1/2^s + 4/10^s).
Multiplicative with a(2^e) = 2^max(0,e-1), a(5^e) = 5^max(0,e-1), a(p^e) = p^e if p = 3 or p >= 7. (End)
a(n) = numerator(n/((n + 2)*(n + 5))).
a(n) = n/b(n), where b(n) = [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, ...] is a purely periodic sequence of period 10. 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 10} A023900(d)*x^d/(1 - x^d)^2 = x/(1 - x)^2 - x^2/(1 - x^2)^2 - 4*x^5/(1 - x^5)^2 + 4*x^10/(1 - x^10)^2.
(End)
Sum_{k=1..n} a(k) ~ (63/200) * n^2. - Amiram Eldar, Nov 25 2022
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MATHEMATICA
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PROG
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(Sage) [lcm(n, 10)/10 for n in range(0, 79)] # Zerinvary Lajos, Jun 07 2009
(GAP) List([0..80], n->NumeratorRat(n/(n+10))); # Muniru A Asiru, Feb 18 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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