OFFSET
0,4
COMMENTS
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
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Wikipedia, Quasi-polynomial.
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).
FORMULA
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
From Peter Bala, Feb 22 2019: (Start)
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)
From Amiram Eldar, Nov 25 2022: (Start)
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)
MAPLE
seq(numer(n/(n+14)), n=0..100); # Nathaniel Johnston, Apr 18 2011
MATHEMATICA
f[n_]:=Numerator[n/(n+14)]; Array[f, 100, 0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)
PROG
(Sage) [lcm(n, 14)/14 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009
(Magma) [Numerator(n/(n+14)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011
(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
CROSSREFS
KEYWORD
nonn,easy,frac,mult
AUTHOR
N. J. A. Sloane, May 15 2005
STATUS
approved