A106620 a(n) = numerator of n/(n+19).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 2, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 3, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• easy
• frac
• mult

Comments

a(n) <> n iff n = 19 * 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 (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-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), A106614 thru A106621 (k = 13 thru 20).

Programs

GAP

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

Magma

[Numerator(n/(n+19)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Maple

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

Mathematica

f[n_]:=Numerator[n/(n+19)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

PARI

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

Sage

[lcm(n,19)/19 for n in range(0, 100)] # Zerinvary Lajos, Jun 12 2009

Formula

Dirichlet g.f.: zeta(s-1)*(1 - 18/19^s). - R. J. Mathar, Apr 18 2011

a(n) = 2*a(n-19) - a(n-38). - G. C. Greubel, Feb 19 2019

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(19^e) = 19^(e-1), and a(p^e) = p^e if p != 19.

Sum_{k=1..n} a(k) ~ (343/722) * n^2. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 37*log(2)/19. - Amiram Eldar, Sep 08 2023