A106614 a(n) = numerator of n/(n+13).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 2, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 3, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 4, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 5, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• frac
• mult
• easy

Comments

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

a(n) <> n iff n = 13 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (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,-1).

Crossrefs

Cf. 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), A106615 thru A106621 (k = 14 thru 20).

Programs

GAP

List([0..80],n->NumeratorRat(n/(n+13))); # Muniru A Asiru, Feb 19 2019

Magma

[Numerator(n/(n+13)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Maple

seq(numer(n/(n+13)),n=0..80); # Muniru A Asiru, Feb 19 2019

Mathematica

f[n_]:=Numerator[n/(n+13)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

PARI

vector(100, n, n--; numerator(n/(n+13))) \\ G. C. Greubel, Feb 19 2019

PARI

a(n)=if(n%13,n,n/13) \\ Charles R Greathouse IV, Jan 24 2022

Sage

[lcm(n,13)/13for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

Formula

G.f.: x/(1-x)^2 - 12*x^13/(1-x^13)^2. - Paul D. Hanna, Jul 27 2005

Dirichlet g.f.: zeta(s-1)*(1-12/13^s). - R. J. Mathar, Apr 18 2011

a(n) = 2*a(n-13) - a(n-26). - G. C. Greubel, Feb 19 2019

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(13^e) = 13^(e-1), and a(p^e) = p^e if p != 13.

Sum_{k=1..n} a(k) ~ (157/338) * n^2. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 25*log(2)/13. - Amiram Eldar, Sep 08 2023