A106615 a(n) = numerator of n/(n+14).

0, 1, 1, 3, 2, 5, 3, 1, 4, 9, 5, 11, 6, 13, 1, 15, 8, 17, 9, 19, 10, 3, 11, 23, 12, 25, 13, 27, 2, 29, 15, 31, 16, 33, 17, 5, 18, 37, 19, 39, 20, 41, 3, 43, 22, 45, 23, 47, 24, 7, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 9, 32, 65, 33, 67, 34, 69, 5, 71, 36, 73, 37, 75, 38, 11, 39

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• easy
• frac
• mult

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).

Crossrefs

Cf. A023900, A106614, A106616, A106621.

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+14))); # Muniru A Asiru, Feb 19 2019

Magma

[Numerator(n/(n+14)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

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 *)

PARI

vector(100, n, n--; numerator(n/(n+14))) \\ G. C. Greubel, Feb 19 2019

Sage

[lcm(n,14)/14 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

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)