A060791 -id:A060791 - cp's OEIS frontend

A132739 Largest divisor of n not divisible by 5.

1, 2, 3, 4, 1, 6, 7, 8, 9, 2, 11, 12, 13, 14, 3, 16, 17, 18, 19, 4, 21, 22, 23, 24, 1, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 36, 37, 38, 39, 8, 41, 42, 43, 44, 9, 46, 47, 48, 49, 2, 51, 52, 53, 54, 11, 56, 57, 58, 59, 12, 61, 62, 63, 64, 13, 66, 67, 68, 69, 14, 71, 72, 73, 74, 3, 76, 77

Offset: 1

Reinhard Zumkeller, Aug 27 2007

A000265(a(n)) = a(A000265(n)) = A132740(n).
a(n) = A060791(n) when n is not divisible by 5. When n is divisible by 5 a(n) divides A060791(n). Tom Edgar, Feb 08 2014
As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (4*5)*G(x^5) - (4*25)*G(x^25) - (4*125)*G(x^125) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (4/5)*H(x^5) - (4/25)*H(x^25) - (4/125)*H(x^125) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (4/5^2)*L(x^5) - (4/25^2)*L(x^25) - (4/125^2)*L(x^125) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} 1/a(n)*x^n = L(x) + (4/5)*L(x^5) + (4/5)*L(x^25) + (4/5)*L(x^125) + ....
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.

Cf. A000265, A038502, A060791, A060904, A112765, A242603.

a132739 n | r > 0     = n
          | otherwise = a132739 n' where (n',r) = divMod n 5
-- Reinhard Zumkeller, Apr 08 2011

f[n_]:=Denominator[5^n/n];Array[f,100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011*)
Table[n/5^IntegerExponent[n, 5], {n, 100}] (* Amiram Eldar, Sep 15 2020 *)

a(n)=n/5^valuation(n, 5) /* Simon Strandgaard, Nov 01 2021 */

def A132739(n):
    a, b = divmod(n, 5)
    while b == 0:
        a, b = divmod(a,5)
    return 5*a+b # Chai Wah Wu, Dec 05 2021

p (1..50).map { |n| n /= 5 while (n % 5) == 0; n } # Simon Strandgaard, Nov 01 2021

a(n) = n/A060904(n). Dirichlet g.f.: zeta(s-1)*(5^s-5)/(5^s-1). - R. J. Mathar, Jul 12 2012
a(n) = n/5^A112765(n). See A060904. - Wolfdieter Lang, Jun 18 2014
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,5^n).
O.g.f.: F(x) - 4*F(x^5) - 4*F(x^25) - 4*F(x^125) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,
Sum_{n >= 0} a(n)^m*x^n = F(m,x) - (5^m - 1)(F(m,x^5) + F(m,x^25) + F(m,x^125) + ...), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)
Sum_{k=1..n} a(k) ~ (5/12) * n^2. - Amiram Eldar, Nov 28 2022

A106609 Numerator of n/(n+8).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 2, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 4, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 6, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 8, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 75, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, May 15 2005

The graph of this sequence is made up of four linear functions: a(n_odd)=n, a(n=2+4i)=n/2, a(4+8i)=n/4, a(8i)=n/8. - Zak Seidov, Oct 30 2006. [In general, f(n) = numerator of n/(n+m) consists of linear functions n/d_i, where d_i are divisors of m (including 1 and m).]
a(n+2), n>=0, is the denominator of the harmonic mean H(n,2) = 4*n/(n+2). a(n+2) = (n+2)/gcd(n+2,8). a(n+5) = A227042(n+2, 2), n >= 0. - Wolfdieter Lang, Jul 04 2013
The sequence p(n) = a(n-4), n>=1, with a(-3) = a(3) = 3, a(-2) = a(2) = 1 and a(-1) = a(1) = 1, appears in the problem of writing 2*sin(2*Pi/n) as an integer in the algebraic number field Q(rho(q(n))), where rho(k) = 2*cos(Pi/k) and q(n) = A225975(n). One has 2*sin(2*Pi/n) = R(p(n), x) modulo C(q(n), x), with x = rho(q(n)) and the integer polynomials R and C given in A127672 and A187360, respectively. See a comment on A225975. - Wolfdieter Lang, Dec 04 2013
A204455(n) divides a(n) for n>=1. - Alexander R. Povolotsky, Apr 06 2015
A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 20 2019

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1).

Cf. A109049, A204455, A225975, A227042 (second column, starting with a(5)).

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).

List([0..80],n->NumeratorRat(n/(n+8))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+8)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

a := n -> iquo(n, [8, 1, 2, 1, 4, 1, 2, 1][1 + modp(n, 8)]):
seq(a(n), n=0..79); # using Wolfdieter Lang's formula, Peter Luschny, Feb 22 2019

f[n_]:=Numerator[n/(n+8)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)
LinearRecurrence[{0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,-1},{0,1,1,3,1,5,3,7,1,9,5,11,3,13,7,15},100] (* Harvey P. Dale, Sep 27 2019 *)

vector(100, n, n--; numerator(n/(n+8))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,8)/8 for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

a(n) = 2*a(n-8) - a(n-16).
G.f.: x* (x^2-x+1) * (x^12 +2*x^11 +4*x^10 +3*x^9 +4*x^8 +4*x^7 +7*x^6 +4*x^5 +4*x^4 +3*x^3 +4*x^2 +2*x +1) / ( (x-1)^2 *(x+1)^2 *(x^2+1)^2 *(x^4+1)^2 ). - R. J. Mathar, Dec 02 2010
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109049(n)/8.
Dirichlet g.f. zeta(s-1)*(1-1/2^s-1/2^(2s)-1/2^(3s)).
Multiplicative with a(2^e) = 2^max(0,e-3). a(p^e) = p^e if p>2. (End)
a(n) = n/gcd(n,8), n >= 0. See the harmonic mean comment above. - Wolfdieter Lang, Jul 04 2013
a(n) = n if n is odd; for n == 0 (mod 8) it is n/8, for n == 2 or 6 (mod 8) it is n/2 and for n == 4 (mod 8) it is n/4. - Wolfdieter Lang, Dec 04 2013
From Peter Bala, Feb 20 2019: (Start)
O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8), where F(x) = x/(1 - x)^2.
More generally, for m >= 1, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m-th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Sum_{n >= 1} (1/n)*a(n)*x^n = G(x) - (1/2)*G(x^2) - (1/4)*G(x^4) - (1/8)*G(x^8), where G(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4)^2*L(x^4) - (1/8)^2*L(x^8), where L(x) = Log(1/(1 - x)).
Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8). (End)
Sum_{k=1..n} a(k) ~ (43/128) * n^2. - Amiram Eldar, Nov 25 2022

A106617 Numerator of n/(n+16).

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 3, 7, 1, 9, 5, 11, 3, 13, 7, 15, 1, 17, 9, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 2, 33, 17, 35, 9, 37, 19, 39, 5, 41, 21, 43, 11, 45, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 15, 61, 31, 63, 4, 65, 33, 67, 17, 69, 35, 71, 9, 73, 37, 75, 19, 77, 39, 79

Offset: 0

N. J. A. Sloane, May 15 2005

A multiplicative sequence. Also, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - 2*G(x^2) - 4*G(x^4) - 8*G(x^8) - 16*G(x^16), where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} n^2*a(n)*x^n = H(x) - 2^2*H(x^2) - 4^2*H(x^4) - 8^2*H(x^8) - 16^2*H(x^16), where H(x) = x*(1 + 4*x + x^2)/(1 - x)^4. In general, the o.g.f. for Sum_{n >= 1} (n^k*a(n))*x^n for positive k involves the Eulerian polynomials.
In the other direction,
Sum_{n >= 1} (a(n)/n)*x^n = J(x) - (1/2)*J(x^2) - (1/4)*J(x^4) - (1/8)*J(x^8) - (1/16)*J(x^16), where J(x) = x/(1 - x).
Sum_{n >= 1} (a(n)/n^2)*x^n = L(x) - (1/2^2)*L(x^2) - (1/4^2)*L(x^4) - (1/8^2)*L(x^8) - (1/16^2)*L(x^16), where L(x) = log(1/(1 - x)). In general, the o.g.f. for Sum_{n >= 0} (a(n)/n^k)*x^n, for k >= 3, involves the polylogarithm Li_(k-1)(x).
Sum_{n >= 1} (1/a(n))*x^n = L(x) + (1/2)*L(x^2) + (1/2)*L(x^4) + (1/2)*L(x^8) + (1/2)*L(x^16). (End)

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (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,-1).

Cf. A106614, A106615, A106616, A106618, A106619, A106620, A106621.

Cf. Other 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).

List([0..100],n->NumeratorRat(n/(n+16))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+16)): n in [0..100]]; // Vincenzo Librandi, Apr 14 2011

seq(numer(n/(n+16)), n=0..100); # Nathaniel Johnston, Apr 18 2011

Numerator/@Table[n/(n+16),{n,0,10}] (* Harvey P. Dale, Apr 26 2011 *)

a(n)=n>>min(valuation(n,2),4) \\ Charles R Greathouse IV, Jun 17 2012

[lcm(n,16)/16 for n in range(0, 100)] # Zerinvary Lajos, Jun 12 2009

a(n) = 2*a(n-16) - a(n-32) for n > 31. - Paul Curtz, Apr 12 2011
Octosections: a(8*n) = A026741(n). a(2+8*n) = 1+4*n. a(4+8*n) = 1+2*n. a(6+8*n) = 3+4*n. Bisection: a(1+2*n) = 1+2*n. - Paul Curtz, Apr 12 2011
Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/4^s-1/8^s-1/16^s). - R. J. Mathar, Apr 18 2011
a(n) = numerator of n/(2^(2*n+1)). - Ralf Steiner, Feb 09 2017
The previous comment is incorrect, a(n) first differs from the numerator of n/(2^(2*n+1)) at n = 32. - Peter Bala, Feb 27 2019
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,16), where gcd(n,16) = [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 16, ...] is a periodic sequence of period 16: a(n) is thus quasi_polynomial in n.
O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4) - F(x^8) - F(x^16), where F(x) = x/(1 - x)^2.
More generally, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 2^m)*( F(m,x^2) + F(m,x^4) + F(m,x^8) + F(m,x^16) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m-th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences ( (n^m)*a(n) )n>=1 for m in Z. Some examples are given below. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^max(0,e-4), and a(p^e) = p^e if p>2.
Sum_{k=1..n} a(k) ~ (171/512) * n^2. (End)

A106619 a(n) = numerator of n/(n+18).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 1, 7, 4, 1, 5, 11, 2, 13, 7, 5, 8, 17, 1, 19, 10, 7, 11, 23, 4, 25, 13, 3, 14, 29, 5, 31, 16, 11, 17, 35, 2, 37, 19, 13, 20, 41, 7, 43, 22, 5, 23, 47, 8, 49, 25, 17, 26, 53, 3, 55, 28, 19, 29, 59, 10, 61, 31, 7, 32, 65, 11, 67, 34, 23, 35, 71, 4, 73, 37, 25, 38, 77, 13, 79

Offset: 0

N. J. A. Sloane, May 15 2005

a(n+3), n >= 0, is the denominator of the harmonic mean H(n,3) = 6*n/(n+3). a(n+3) = (n+3)/gcd(n+3,18). - Wolfdieter Lang, Jul 04 2013

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

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).

Cf. A227042.

List([0..80],n->NumeratorRat(n/(n+18))); # Muniru A Asiru, Feb 19 2019

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

seq(numer(n/(n+18)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+18)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

vector(100, n, n--; numerator(n/(n+18))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,18)/18 for n in range(0, 100)] # Zerinvary Lajos, Jun 12 2009

a(n) = 2*a(n-18) - a(n-36). - Paul Curtz, Feb 27 2011
Nonasection: a(9*n) = A026741(n). - Paul Curtz, Mar 21 2011
Dirichlet g.f.: zeta(s-1)*(1 - 2/3^s - 2/9^s - 1/2^s + 2/6^s + 2/18^s). - R. J. Mathar, Apr 18 2011
a(n) = n/gcd(n,18), n >= 0. See the harmonic mean comment above, and the Zerinvary Lajos program below. - Wolfdieter Lang, Jul 04 2013
a(n+3) = A227042(n+3,3), n >= 0. - Wolfdieter Lang, Jul 04 2013
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(2^e) = 2^max(0, e-1), a(3^e) = 3^max(0,e-2), and a(p^e) = p^e otherwise.
Sum_{k=1..n} a(k) ~ (61/216) * n^2. (End)

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

Original entry on oeis.org

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

N. J. A. Sloane, May 15 2005

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

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).

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).

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

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

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

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

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

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

Original entry on oeis.org

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

N. J. A. Sloane, May 15 2005

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

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).

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).

List([0..80],n->NumeratorRat(n/(n+14))); # Muniru A Asiru, Feb 19 2019

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

seq(numer(n/(n+14)),n=0..100); # Nathaniel Johnston, Apr 18 2011

f[n_]:=Numerator[n/(n+14)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

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

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

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)

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

Original entry on oeis.org

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

N. J. A. Sloane, May 15 2005

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

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).

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).

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

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

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

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

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

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

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

A106610 Numerator of n/(n+9).

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 19, 20, 7, 22, 23, 8, 25, 26, 3, 28, 29, 10, 31, 32, 11, 34, 35, 4, 37, 38, 13, 40, 41, 14, 43, 44, 5, 46, 47, 16, 49, 50, 17, 52, 53, 6, 55, 56, 19, 58, 59, 20, 61, 62, 7, 64, 65, 22, 67, 68, 23, 70, 71, 8, 73, 74, 25, 76, 77

Offset: 0

N. J. A. Sloane, May 15 2005

Apart from 0, also numerator of Sum_{i=1..n} (1/((i+2)*(i+3))) = n/(3n+9). - Bruno Berselli, Nov 07 2012

In addition to being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n >= 1, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

			For n = 12, n/(n+9) = 12/21 = 4/7. So, a(12) = 4. - _Indranil Ghosh_, Jan 31 2017
From _Peter Bala_, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (2*3)*G(x^3) - (2*9)*G(x^9) , where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (2/3)*H(x^3) - (2/9)*H(x^9), where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (2/3^2)*L(x^3) - (2/9^2)*L(x^9), where L(x) = Log(1/(1 - x)).
Sum_{n >= 1} (1/a(n))*x^n = L(x) + (2/3)*L(x^3) + (2/3)*L(x^9). (End)

Raffaello Giusti, editore, Supplemento al Periodico di Matematica (Livorno), Jul 1902, p. 138 (Problem 421, case k=3).

Indranil Ghosh, Table of n, a(n) for n = 0..20000 (terms 0..1000 from Vincenzo Librandi)
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0, 0,-1).

Cf. A008292, A109050.

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).

List([0..80],n->NumeratorRat(n/(n+9))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+9)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+9)),n=0..80); # Muniru A Asiru, Feb 19 2019

f[n_]:=Numerator[n/(n+9)]; Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)

vector(100, n, n--; numerator(n/(n+9))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,9)/9for n in range(0, 100)] # Zerinvary Lajos, Jun 09 2009

From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109050(n)/9.
Dirichlet g.f. zeta(s-1)*(1-2/3^s-2/9^s).
Multiplicative with a(3^e) = 3^max(0,e-2), a(p^e) = p^e if p<>3. (End)
a(n) = 2*a(n-9) - a(n-18). - G. C. Greubel, Feb 19 2019
From Peter Bala, Feb 21 2019: (Start)
a(n) = n/gcd(n,9), where gcd(n,9) = [1, 1, 3, 1, 1, 3, 1, 1, 9, ...] is a periodic sequence of period 9: a(n) is thus quasi_polynomial in n.
O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - 2*F(x^3) - 2*F(x^9), where F(x) = x/(1 - x)^2.
More generally, for m >= 1, Sum_{n >= 0} (a(n)^m)*x^n = F(m,x) + (1 - 3^m)*( F(m,x^3) + F(m,x^9) ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.
Repeatedly applying the Euler operator x*d/dx or its inverse to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)
Sum_{k=1..n} a(k) ~ (61/162) * n^2. - Amiram Eldar, Nov 25 2022

A106618 a(n) = numerator of n/(n+17).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 2, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 3, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 4, 69, 70, 71, 72, 73, 74

Offset: 0

N. J. A. Sloane, May 15 2005

a(n) <> n iff n = 17 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1).

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).

List([0..80],n->NumeratorRat(n/(n+17))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+17)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+17)),n=0..80); # Muniru A Asiru, Feb 19 2019

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

vector(100, n, n--; numerator(n/(n+17))) \\ G. C. Greubel, Feb 19 2019

[lcm(n,17)/17 for n in range(0, 100)] # Zerinvary Lajos, Jun 12 2009

Dirichlet g.f.: zeta(s-1)*(1 - 16/17^s). - R. J. Mathar, Apr 18 2011
a(n) = 2*a(n-17) - a(n-34). - G. C. Greubel, Feb 19 2019
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(17^e) = 17^(e-1), and a(p^e) = p^e if p != 17.
Sum_{k=1..n} a(k) ~ (273/578) * n^2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 33*log(2)/17. - Amiram Eldar, Sep 08 2023

A109046 a(n) = lcm(n, 5).

Original entry on oeis.org

0, 5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 55, 60, 65, 70, 15, 80, 85, 90, 95, 20, 105, 110, 115, 120, 25, 130, 135, 140, 145, 30, 155, 160, 165, 170, 35, 180, 185, 190, 195, 40, 205, 210, 215, 220, 45, 230, 235, 240, 245, 50, 255, 260, 265, 270, 55, 280, 285, 290

Offset: 0

Mitch Harris, Jun 18 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,2,0,0,0,0,-1).
Index entries for sequences related to lcm's.

Cf. A060791, A109009, A109042.

[Lcm(n,5): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

a[n_] := LCM[n, 5]; Array[a, 60, 0] (* Amiram Eldar, Nov 26 2022 *)
LinearRecurrence[{0,0,0,0,2,0,0,0,0,-1},{0,5,10,15,20,5,30,35,40,45},60] (* Harvey P. Dale, Oct 08 2023 *)

[lcm(n,5)for n in range(0, 59)] # Zerinvary Lajos, Jun 07 2009

a(n) = n*5/gcd(n, 5) = 5*n/A109009(n) = 5*A060791(n).
G.f.: 5*x*(x^4+x^3+3*x^2+x+1)*(x^4+x^3-x^2+x+1) / ( (x-1)^2*(x^4+x^3+x^2+x+1)^2 ). - R. J. Mathar, Apr 18 2011
Sum_{k=1..n} a(k) ~ (21/10) * n^2. - Amiram Eldar, Nov 26 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = 9*log(2)/25. - Amiram Eldar, Sep 08 2023

cp's OEIS Frontend

A132739 Largest divisor of n not divisible by 5.

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A106609 Numerator of n/(n+8).

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A106617 Numerator of n/(n+16).

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A106619 a(n) = numerator of n/(n+18).

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

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

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