A060791 -id:A060791 - cp's OEIS frontend

A106616 Numerator of n/(n+15).

0, 1, 2, 1, 4, 1, 2, 7, 8, 3, 2, 11, 4, 13, 14, 1, 16, 17, 6, 19, 4, 7, 22, 23, 8, 5, 26, 9, 28, 29, 2, 31, 32, 11, 34, 7, 12, 37, 38, 13, 8, 41, 14, 43, 44, 3, 46, 47, 16, 49, 10, 17, 52, 53, 18, 11, 56, 19, 58, 59, 4, 61, 62, 21, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26

Offset: 0

N. J. A. Sloane, May 15 2005

Multiplicative and also a strong divisibility sequence: gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Feb 24 2019

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

Cf. A000010, A106614, A106615, A106617, 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..80],n->NumeratorRat(n/(n+15))); # Muniru A Asiru, Feb 19 2019

[Numerator(n/(n+15)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

seq(numer(n/(n+15)),n=0..100); # Nathaniel Johnston, Apr 18 2011

f[n_]:=Numerator[n/(n+15)]; Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011*)

a(n) = numerator(n/(n+15)); \\ Michel Marcus, Feb 19 2017

[lcm(n,15)/15 for n in range(0, 79)] # Zerinvary Lajos, Jun 09 2009

Dirichlet g.f.: zeta(s-1)*(1-4/5^s-2/3^s+8/15^s). - R. J. Mathar, Apr 18 2011
a(n) = gcd((n-2)*(n-1)*n*(n+1)*(n+2)/15, n) for n>=1. - Lechoslaw Ratajczak, Feb 19 2017
From Peter Bala, Feb 24 2019: (Start)
a(n) = n/gcd(n,15), a quasi-polynomial in n since gcd(n,15) is a purely periodic sequence of period 15.
O.g.f.: F(x) - 2*F(x^3) - 4*F(x^5) + 8*F(x^15), where F(x) = x/(1 - x)^2.
O.g.f. for reciprocals: Sum_{n >= 1} x^n/a(n) = Sum_{d divides 15} (phi(d)/d) * log(1/(1 - x^d)) = log(1/(1 - x)) + (2/3)*log(1/(1 - x^3)) + (4/5)*log(1/(1 - x^5)) + (8/15)*log(1/(1 - x^15)), where phi(n) denotes the Euler totient function A000010. (End)
From Amiram Eldar, Nov 25 2022: (Start)
Multiplicative with a(3^e) = 3^max(0,e-1), a(5^e) = 5^max(0,e-1), and a(p^e) = p^e otherwise.
Sum_{k=1..n} a(k) ~ (49/150) * n^2. (End)

A175784 Numerators of k/(10+k)+1 for k = 2*n-1.

Original entry on oeis.org

12, 16, 4, 24, 28, 32, 36, 8, 44, 48, 52, 56, 12, 64, 68, 72, 76, 16, 84, 88, 92, 96, 20, 104, 108, 112, 116, 24, 124, 128, 132, 136, 28, 144, 148, 152, 156, 32, 164, 168, 172, 176, 36, 184, 188, 192, 196, 40, 204, 208, 212, 216, 44, 224, 228

Offset: 1

Steven J. Forsberg, Dec 04 2010

nonn

For even k the expression k/(k+10)+1 yields A060791 as denominators, A096431 as numerators. For odd k it yields A096431 as denominators, the present sequence as numerators.

Note that A096431 is denominator of (9*(n^4 - 2n^3 + 2n^2 - n) + 2)/(2*(2*n-1)), equivalently denominator of (3*n^2 - 3*n + 1)*(3*n^2 - 3*n + 2)/(2*n-1), and that A060791 is n/gcd(n,5).

			n=1: (2*1-1)/(2*1+9)+1 = 1/11+1 = 12/11, hence a(1) = 12;
n=2: (2*2-1)/(2*2+9)+1 = 3/13+1 = 16/13, hence a(2) = 16;
n=3: (2*3-1)/(2*3+9)+1 = 5/15+1 = 1/3+1 = 4/3, hence a(3) = 4;

Cf. A096431, A060791.

A175784 := proc(n) local k ; k := 2*n-1 ; numer(k/(10+k)+1) ; end proc:
seq(A175784(n),n=1..30) ; # R. J. Mathar, Feb 05 2011

Numerator[Table[k/(10 + k) + 1, {k, 1, 100, 2}]]

a(n) = numerator((2*n-1)/(2*n+9) + 1).

Conjecture: a(n) = 2*a(n-5) - a(n-10) = 4*A060791(n+2) with g.f. -4*x*(-3 - 4*x - x^2 - 6*x^3 - 7*x^4 - 2*x^5 - x^6 + x^8 + 2*x^9) / ( (x-1)^2*(x^4 + x^3 + x^2 + x + 1)^2 ). [R. J. Mathar, Dec 07 2010]

cp's OEIS Frontend

A106616 Numerator of n/(n+15).

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