A106608 a(n) = numerator of n/(n+7).

0, 1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 11, 12, 13, 2, 15, 16, 17, 18, 19, 20, 3, 22, 23, 24, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 5, 36, 37, 38, 39, 40, 41, 6, 43, 44, 45, 46, 47, 48, 7, 50, 51, 52, 53, 54, 55, 8, 57, 58, 59, 60, 61, 62, 9, 64, 65, 66, 67, 68, 69, 10, 71, 72, 73, 74, 75, 76

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• easy
• frac
• mult

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

Generalizing the above comment of Hanna, the numerators of n/(n + k) for a fixed positive integer k and n >= 0 form a sequence with the g.f.: Sum_{d divides k} f(d)*x^d/(1 - x^d)^2, where f(n) is the Dirichlet inverse of the Euler totient function A000010. f(n) is a multiplicative function defined on prime powers p^k by f(p^k) = 1 - p. See A023900. - Peter Bala, Feb 17 2019

a(n) <> n iff n = 7 * 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 (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).

Crossrefs

Cf. A000010, A109048, A023900.

Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106609 thru A106612 (k = 8 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Programs

GAP

List([0..80],n->NumeratorRat(n/(n+7))); # Muniru A Asiru, Feb 19 2019

Magma

[Numerator(n/(n+7)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Maple

seq(numer(n/(n+7)),n=0..80); # Muniru A Asiru, Feb 19 2019

Mathematica

f[n_]:=Numerator[n/(n+7)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)

PARI

vector(100, n, n--; numerator(n/(n+7))) \\ G. C. Greubel, Feb 19 2019

Sage

[lcm(n,7)/7 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

Formula

G.f.: x/(1-x)^2 - 6*x^7/(1-x^7)^2. - Paul D. Hanna, Jul 27 2005

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

a(n) = A109048(n)/7.

Dirichlet g.f.: zeta(s-1)*(1-6/7^s).

Multiplicative with a(p^e) = p^e if p<>7, a(7^e) = 7^(e-1) if e > 0. (End)

Sum_{k=1..n} a(k) ~ (43/98) * n^2. - Amiram Eldar, Nov 25 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = 13*log(2)/7. - Amiram Eldar, Sep 08 2023