A106611 a(n) = numerator of n/(n+10).

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

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• easy
• frac
• mult

Comments

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..5000
• Wikipedia, Quasi-polynomial.
• 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).

Crossrefs

Cf. A023900, A109051, A303367.

Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Programs

GAP

List([0..80],n->NumeratorRat(n/(n+10))); # Muniru A Asiru, Feb 18 2019

Magma

[Numerator(n/(n+10)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Mathematica

f[n_]:=Numerator[n/(n+10)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

Sage

[lcm(n,10)/10 for n in range(0, 79)] # Zerinvary Lajos, Jun 07 2009

Formula

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109051(n)/10.

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)

From Peter Bala, Feb 17 2019: (Start)

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