A109008 -id:A109008 - cp's OEIS frontend

A060819 a(n) = n / gcd(n,4).

1, 1, 3, 1, 5, 3, 7, 2, 9, 5, 11, 3, 13, 7, 15, 4, 17, 9, 19, 5, 21, 11, 23, 6, 25, 13, 27, 7, 29, 15, 31, 8, 33, 17, 35, 9, 37, 19, 39, 10, 41, 21, 43, 11, 45, 23, 47, 12, 49, 25, 51, 13, 53, 27, 55, 14, 57, 29, 59, 15, 61, 31, 63, 16, 65, 33, 67, 17, 69, 35, 71, 18, 73, 37, 75, 19

Offset: 1

Len Smiley, Apr 30 2001

From Peter Bala, Feb 19 2019: (Start)
We make some general remarks about the sequence a(n) = numerator(n/(n + k)) = n/gcd(n,k) for k a fixed positive integer. The present sequence is the case k = 4. Several other cases are listed in the Crossrefs. In addition to being multiplicative these sequences are also strong divisibility sequences, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m).
By the multiplicativeness and strong divisibility property of the sequence a(n) it follows that 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.
The sequence a(n) has the rational generating function 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. Cf. A181318. (End)

			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), where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (1/2)*H(x^2) - (1/4)*H(x^4), where H(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), 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). (End)

Harry J. Smith, Table of n, a(n) for n = 1..1000
Peter Bala, A note on the sequence of numerators of a rational function, 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A026741, A051176, A060791, A060789. Cf. Other sequences given by the formula numerator(n/(n + k)): A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Cf. A000010, A023900, A061037, A061038, A109008, A145979, A167192, A176895, A181318, A220466.

List([1..80],n->n/Gcd(n,4)); # Muniru A Asiru, Feb 20 2019

a060819 n = n `div` a109008 n  -- Reinhard Zumkeller, Nov 25 2013

[n/GCD(n,4): n in [1..80]]; // G. C. Greubel, Sep 19 2018

A060819 := n -> numer(1/2-1/(n+2)): seq(A060819(n),n=1..75); # Gary Detlefs, Sep 16 2011

Mathematica
```
f[n_]:= n/GCD[n, 4]; Array[f, 80]
```

a(n) = { n / gcd(n, 4) } \\ Harry J. Smith, Jul 12 2009

[lcm(n,4)/4 for n in (1..80)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 +x +3*x^2 +x^3 +3*x^4 +x^5 +x^6)/(1 - x^4)^2.
a(n) = 2*a(n-4) - a(n-8).
a(n) = (n/16)*(11 - 5*(-1)^n - i^n - (-i)^n). - Ralf Stephan, Mar 15 2003
a(2*n+1) = a(4*n+2) = 2*n+1, a(4*n+4) = n+1. - Ralf Stephan, Jun 10 2005
Multiplicative with a(2^e) = 2^max(0, e-2), a(p^e) = p^e, p >= 3. - Mitch Harris, Jun 29 2005
a(n) = A167192(n+4,4). - Reinhard Zumkeller, Oct 30 2009
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109045(n)/4.
Dirichlet g.f.: zeta(s-1)*(1-1/2^s-1/2^(2s)). (End)
a(n+4) - a(n) = A176895(n).  - Paul Curtz, Apr 05 2011
a(n) = numerator(Sum_{k=1..n} 1/((k+1)*(k+2))). This summation has a closed form of 1/2 - 1/(n+2) and denominator of A145979(n). - Gary Detlefs, Sep 16 2011
a((2*n-1)*2^p) = ceiling(2^(p-2))*(2*n-1), p >= 0 and n >= 1. - Johannes W. Meijer, Feb 06 2013
a(n) = n / A109008(n). - Reinhard Zumkeller, Nov 25 2013
a(n) = denominator((2n-4)/n). - Wesley Ivan Hurt, Dec 22 2016
From Peter Bala, Feb 21 2019: (Start)
O.g.f.: Sum_{n >= 0} a(n)*x^n = F(x) - F(x^2) - F(x^4), 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) ), 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 a(n) produces generating functions for the sequences ((n^m)*a(n))n>=1 for m in Z. Some examples are given below.
(End)
Sum_{k=1..n} a(k) ~ (11/32) * n^2. - Amiram Eldar, Nov 25 2022
E.g.f.: x*(8*cosh(x) + sin(x) + 3*sinh(x))/8. - Stefano Spezia, Dec 02 2023

A003989 Triangle T from the array A(x, y) = gcd(x,y), for x >= 1, y >= 1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1, 1, 2, 1

Offset: 1

Marc LeBrun

For m < n, the maximal number of nonattacking queens that can be placed on the n by m rectangular toroidal chessboard is gcd(m,n), except in the case m=3, n=6.
The determinant of the matrix of the first n rows and columns is A001088(n). [Smith, Mansion] - Michael Somos, Jun 25 2012
Imagine a torus having regular polygonal cross-section of m sides. Now, break the torus and twist the free ends, preserving rotational symmetry, then reattach the ends. Let n be the number of faces passed in twisting the torus before reattaching it. For example, if n = m, then the torus has exactly one full twist. Do this for arbitrary m and n (m > 1, n > 0). Now, count the independent, closed paths on the surface of the resulting torus, where a path is "closed" if and only if it returns to its starting point after a finite number of times around the surface of the torus. Conjecture: this number is always gcd(m,n). NOTE: This figure constitutes a group with m and n the binary arguments and gcd(m,n) the resulting value. Twisting in the reverse direction is the inverse operation, and breaking & reattaching in place is the identity operation. - Jason Richardson-White, May 06 2013
Regarded as a triangle, table of gcd(n - k +1, k) for 1 <= k <= n. - Franklin T. Adams-Watters, Oct 09 2014
The n-th row of the triangle is 1,...,1, if and only if, n + 1 is prime. - Alexandra Hercilia Pereira Silva, Oct 03 2020

			The array A begins:
  [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
  [1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2]
  [1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1]
  [1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4]
  [1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1]
  [1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]
  [1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]
  [1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]
  [1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]
  [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]
  [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]
  [1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]
  ...
The triangle T begins:
  n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  1:  1
  2:  1  1
  3:  1  2  1
  4:  1  1  1  1
  5:  1  2  3  2  1
  6:  1  1  1  1  1  1
  7:  1  2  1  4  1  2  1
  8:  1  1  3  1  1  3  1  1
  9:  1  2  1  2  5  2  1  2  1
 10:  1  1  1  1  1  1  1  1  1  1
 11:  1  2  3  4  1  6  1  4  3  2  1
 12:  1  1  1  1  1  1  1  1  1  1  1  1
 13:  1  2  1  2  1  2  7  2  1  2  1  2  1
 14:  1  1  3  1  5  3  1  1  3  5  1  3  1  1
 15:  1  2  1  4  1  2  1  8  1  2  1  4  1  2  1
 ...  - _Wolfdieter Lang_, May 12 2018

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, ch. 4.
D. E. Knuth, The Art of Computer Programming, Addison-Wesley, section 4.5.2.

T. D. Noe, First 100 antidiagonals of array, flattened
Grant Cairns, Queens on Non-square Tori, El. J. Combinatorics, N6, 2001.
Marc Chamberland, Factored matrices can generate combinatorial identities, Linear Algebra and its Applications, Volume 438, Issue 4, 2013, pp. 1667-1677.
Warren P. Johnson, An LDU Factorization in Elementary Number Theory, Mathematics Magazine, Vol. 76, No. 5 (Dec., 2003), pp. 392-394.
P. Mansion, On an Arithmetical Theorem of Professor Smith's, Messenger of Mathematics, (1878), pp. 81-82.
Kival Ngaokrajang, Pattern of GCD(x,y) > 1 for x and y = 1..60. Non-isolated values larger than 1 (polyomino shapes) are colored.
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.
Mihai Prunescu and Joseph Shunia, Arithmetic-term representations for the greatest common divisor, arXiv:2411.06430 [math.NT], 2024.
H. J. S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875-1876), pp. 208-212.
Index entries for sequences related to lcm's

Rows, columns and diagonals: A089128, A109007, A109008, A109009, A109010, A109011, A109012, A109013, A109014, A109015.
A109004 is (0, 0) based.
Cf.  A001088, A003990, A003991, A050873, A051731, A054431, A054522.
Cf. also A091255 for GF(2)[X] polynomial analog.
A(x, y) = A075174(A004198(A075173(x), A075173(y))) = A075176(A004198(A075175(x), A075175(y))).
Antidiagonal sums are in A006579.

Flat(List([1..15],n->List([1..n],k->Gcd(n-k+1,k)))); # Muniru A Asiru, Aug 26 2018

a:=(n,k)->gcd(n-k+1,k): seq(seq(a(n,k),k=1..n),n=1..15); # Muniru A Asiru, Aug 26 2018

Table[ GCD[x - y + 1, y], {x, 1, 15}, {y, 1, x}] // Flatten (* Jean-François Alcover, Dec 12 2012 *)

{A(n, m) = gcd(n, m)}; /* Michael Somos, Jun 25 2012 */

Multiplicative in both parameters with a(p^e, m) = gcd(p^e, m). - David W. Wilson, Jun 12 2005
T(n, k) = A(n - k + 1, k) = gcd(n - k + 1, k), n >= 1, k = 1..n. See a comment above and the Mathematica program. - Wolfdieter Lang, May 12 2018
Dirichlet generating function: Sum_{n>=1} Sum_{k>=1} gcd(n, k)/n^s/k^c = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(s + c). - Mats Granvik, Feb 13 2021
The LU decomposition of this square array = A051731 * transpose(A054522) (see Johnson (2003) or Chamberland (2013), p. 1673). - Peter Bala, Oct 15 2023

A109004 Table of gcd(n, m) read by antidiagonals, n >= 0, m >= 0.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 1, 2, 1, 4, 5, 1, 1, 1, 1, 5, 6, 1, 2, 3, 2, 1, 6, 7, 1, 1, 1, 1, 1, 1, 7, 8, 1, 2, 1, 4, 1, 2, 1, 8, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13

Offset: 0

Mitch Harris

			Triangle starts:
  [ 0] [0]
  [ 1] [1, 1]
  [ 2] [2, 1, 2]
  [ 3] [3, 1, 1, 3]
  [ 4] [4, 1, 2, 1, 4]
  [ 5] [5, 1, 1, 1, 1, 5]
  [ 6] [6, 1, 2, 3, 2, 1, 6]
  [ 7] [7, 1, 1, 1, 1, 1, 1, 7]
  [ 8] [8, 1, 2, 1, 4, 1, 2, 1, 8]
  [ 9] [9, 1, 1, 3, 1, 1, 3, 1, 1, 9]
  [10] [10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10]
  [11] [11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11]
  [12] [12, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12]

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 335.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, ch. 4.
D. E. Knuth, The Art of Computer Programming, Addison-Wesley, section 4. 5. 2

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Peter Luschny, The binary GCD algorithm, a Python implementation.
Marcelo Polezzi, A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor, The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), pp. 445-446.

Rows, columns and diagonals: A089128, A109007, A109008, A109009, A109010, A109011, A109012, A109013, A109014, A109015.

A003989 is (1, 1) based.

a[n_, m_] := GCD[n, m]; Table[a[n - m, m], {n,0,10}, {m,0,n}]//Flatten (* G. C. Greubel, Jan 04 2018 *)

PARI
```
{a(n, m) = gcd( n, m)}
```

{a(n, m) = local(x); n = abs(n); m = abs(m); if( !m, n, -2 * sum( k=1, m, x = k * n / m; x - floor( x) - 1/2))} /* Michael Somos, May 22 2011 */

# Since 3.5 part of the math module. For a version using the binary GCD algorithm see the links.
for n in range(13): print([math.gcd(n, k) for k in range(n + 1)])  # Peter Luschny, May 14 2025

a(n, m) = a(m, n) = a(m, n-m) = a(m, n mod m), n >= m.
a(n, m) = n + m - n*m + 2*Sum_{k=1..m-1} floor(k*n/m).
Multiplicative in both parameters with a(p^e, m) = gcd(p^e, m). - David W. Wilson, Jun 12 2005

A010121 Continued fraction for sqrt(7).

Original entry on oeis.org

2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4

Offset: 0

N. J. A. Sloane

This is a basic member of a family of 4-periodic multiplicative sequences with two parameters (c1,c2), defined for n >= 1 by a(n)=1 if n is odd, a(n)=c1 if n == 0 (mod 4) and a(n)=c2 if n == 2 (mod 4). Here, (c1,c2)=(4,1).
The Dirichlet generating function is (1+(c2-1)/2^s+(c1-c2)/4^s)*zeta(s).
Other members are A010123 with parameters (6,2), A010127 (8,3), A010130 (10,1), A010131 (10,2), A010132 (10,4), A010137 (12,5), A010146 (14,6), A089146 (4,8), A109008 (4,2), A112132 (7,3). If c1=c2, this reduces to the cases discussed in A040001. - R. J. Mathar, Feb 18 2011

			2.645751311064590590501615753...  = A010465 = 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))).

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4, example 5.
G. Xiao, Contfrac
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010465 (decimal expansion).

ContinuedFraction[Sqrt[7],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
CoefficientList[Series[(2 x^2 + 3 x + 2) (x^2 - x + 1) / ((1 - x) (1 + x) (x^2 + 1)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 26 2016 *)
PadRight[{2},120,{4,1,1,1}] (* Harvey P. Dale, Nov 30 2019 *)

{ allocatemem(932245000); default(realprecision, 13000); x=contfrac(sqrt(7)); for (n=0, 20000, write("b010121.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

From R. J. Mathar, Jun 17 2009: (Start)
G.f.: -(2*x^2+3*x+2)*(x^2-x+1)/((x-1)*(1+x)*(x^2+1)).
a(n) = a(n-4), n > 4. (End)
a(n) = (7 + 3*(-1)^n + 3*(-i)^n + 3*i^n)/4, n > 0, where i is the imaginary unit. - Bruno Berselli, Feb 18 2011

A109012 a(n) = gcd(n,9).

Original entry on oeis.org

9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1

Offset: 0

Mitch Harris

Start with positive integer n. At each step, either (a) multiply by any positive integer or (b) remove all zeros from the number. a(n) is the smallest number that can be reached by this process. - David W. Wilson, Nov 01 2005
From Martin Fuller, Jul 09 2007: (Start)
Also the minimal positive difference between numbers whose digit sum is a multiple of n. Proof:
Construction: Pick a positive number that does not end with 9, and has a digit sum n-a(n). To form the lower number, append 9 until the digit sum is a multiple of n. This is always possible since the difference is gcd(n,9). Add a(n) to form the higher number, which will have digit sum n.
E.g., n=12: prefix=18, lower=18999, higher=19002, difference=3.
Minimality: All numbers are a multiple of a(n) if their digit sum is a multiple of n. Hence the minimal difference is at least a(n). (End)

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A109004.
Cf. A109007, A109008, A109009, A109010, A109011, A109013, A109014, A109015.
Cf. A074232, A374039.

GCD[Range[0, 100], 9] (* Paolo Xausa, Aug 28 2024 *)

a(n)=gcd(n,9) \\ Charles R Greathouse IV, Oct 07 2015

from math import gcd
def a(n): return gcd(n, 9)
print([a(n) for n in range(101)]) # Michael S. Branicky, Sep 01 2021

a(n) = 1 + 2*[3|n] + 6*[9|n], where [x|y] = 1 when x divides y, 0 otherwise.
a(n) = a(n-9).
Multiplicative with a(p^e, 9) = gcd(p^e, 9). - David W. Wilson, Jun 12 2005
G.f.: (-9 - x - x^2 - 3*x^3 - x^4 - x^5 - 3*x^6 - x^7 - x^8) / ((x-1)*(1 + x + x^2)*(x^6 + x^3 + 1)). - R. J. Mathar, Apr 04 2011
Dirichlet g.f.: (1+2/3^s+6/9^s)*zeta(s). - R. J. Mathar, Apr 04 2011

A146160 Period 4: repeat [1, 4, 1, 16].

Original entry on oeis.org

1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1

Offset: 1

Artur Jasinski, Oct 27 2008

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010156, A145996. [Artur Jasinski, Oct 29 2008]

&cat[[1,4,1,16]^^20]; // Vincenzo Librandi, Feb 04 2016

A146160:=n->[1, 4, 1, 16][(n mod 4)+1]: seq(A146160(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016

Table[GCD[4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2], {k, 100}]
PadRight[{},100,{1,4,1,16}] (* or *) LinearRecurrence[{0,0,0,1},{1,4,1,16},90] (* Harvey P. Dale, Mar 29 2025 *)

Vec((1+4*x+x^2+16*x^3)/(1-x^4) + O(x^100)) \\ Altug Alkan, Feb 04 2016

Continued fraction of (8 + sqrt(78))/14.
GCD(4k - k^2, 5k^2, 20k - 20k^2, 16 - 32k + 16k^2) for k = 1,2,3,...
From Artur Jasinski, Oct 29 2008: (Start)
a(n) = 1 when n congruent to 1 or 3 mod 4.
a(n) = 4 when n congruent to 2 mod 4.
a(n) = 16 when n congruent to 0 mod 4. (End)
From Richard Choulet, Nov 03 2008: (Start)
a(n+4) = a(n).
a(n) = (9/2)*(-1)^n + (11/2) + 6*cos(Pi*n/2).
O.g.f.: f(z) = a(0)+a(1)*z+... = (1+4*z+z^2+16*z^3)/(1-z^4). (End)
E.g.f.: sinh(x) + 20*(sinh(x/2))^2 - 12*(sin(x/2))^2. - G. C. Greubel, Feb 03 2016
a(n) = a(-n). - Wesley Ivan Hurt, Jun 15 2016
a(n) = A109008(n)^2. - R. J. Mathar, Feb 12 2019
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(12/4^s+3/2^s+1). (End)

Choulet formula adapted for offset 1 from Wesley Ivan Hurt, Jun 15 2016

A109045 a(n) = lcm(n,4).

Original entry on oeis.org

0, 4, 4, 12, 4, 20, 12, 28, 8, 36, 20, 44, 12, 52, 28, 60, 16, 68, 36, 76, 20, 84, 44, 92, 24, 100, 52, 108, 28, 116, 60, 124, 32, 132, 68, 140, 36, 148, 76, 156, 40, 164, 84, 172, 44, 180, 92, 188, 48, 196, 100, 204, 52, 212, 108, 220, 56, 228, 116, 236, 60

Offset: 0

Mitch Harris, Jun 18 2005

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

Cf. A060819, A084351, A109008, A109042.

a109045 = lcm 4  -- Reinhard Zumkeller, Nov 25 2013

[Lcm(n,4): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Table[LCM[n,4],{n,0,80}] (* Harvey P. Dale, May 02 2011 *)

a(n) = lcm(n, 4); \\ Michel Marcus, Mar 07 2018

a(n) = 4n/gcd(n, 4).
a(n) = A084351(n), n > 1. - R. J. Mathar, Aug 20 2008
From R. J. Mathar, Apr 18 2011: (Start)
G.f.: 4*x*(1+x+3*x^2+x^3+3*x^4+x^5+x^6) / ( (x-1)^2*(1+x)^2*(x^2+1)^2 ).
a(n) = 4*n/A109008(n) = 4*A060819(n). (End)
Sum_{k=1..n} a(k) ~ (11/8) * n^2. - Amiram Eldar, Nov 26 2022

A247004 Denominator of (n+4)/gcd(n, 4)^2, a 16-periodic sequence that associates A061037 with A106617.

Original entry on oeis.org

4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4, 1, 2, 1, 2

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 09 2014

This sequence may also be defined as the denominators of A061037(n+3)/(n+1), or also as A060819 / A109008.

One can notice that the analog numerators [numerators of (n+4)/gcd(n, 4)^2] are A106617 left-shifted 4 places.

			Fractions begin:
1/4,  5,  3/2,  7, 1/2,  9,  5/2, 11, 3/4, 13,  7/2, 15, 1, 17,  9/2, 19,
5/4, 21, 11/2, 23, 3/2, 25, 13/2, 27, 7/4, 29, 15/2, 31, 2, 33, 17/2, 35,
...
Numerators begin:
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,
...
Periodic part = [4, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1];

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A060819, A061037, A106617, A109008, A188134, A213268.

[Denominator((n+4)/Gcd(n,4)^2): n in [0..100]]; // G. C. Greubel, Aug 05 2018

a[n_] := (n+4)/GCD[n, 4]^2 // Denominator;  Table[a[n], {n, 0, 100}]
(* or: *)
Table[{1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 1, 2, 1, 4}[[Mod[n, 16, 1]]], {n, 0, 100}]

for(n=0,100, print1(denominator((n+4)/gcd(n,4)^2), ", ")) \\ G. C. Greubel, Aug 05 2018

A247004 = A060819 / A109008.
(n+4) / gcd(n, 4)^2 = A188134(n+4) / 4. - Michael Somos, Sep 12 2014
a(n) = a(n+16) = a(-n), a(2*n + 1) = 1 for all n in Z. - Michael Somos, Sep 13 2014

A251091 a(n) = n^2 / gcd(n+2, 4).

Original entry on oeis.org

0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121, 72, 169, 49, 225, 128, 289, 81, 361, 200, 441, 121, 529, 288, 625, 169, 729, 392, 841, 225, 961, 512, 1089, 289, 1225, 648, 1369, 361, 1521, 800, 1681, 441, 1849, 968, 2025, 529, 2209, 1152, 2401, 625, 2601, 1352

Offset: 0

Paul Curtz, May 08 2015

A061038(n), which appears in 4*a(n) formula, is a permutation of n^2.
Origin. In December 2010, I wrote in my 192-page Exercise Book no. 5, page 41, the array (difference table of the first row):
   1     0,   1/3,     1,   9/5,    8/3,   25/7,    9/2,   49/9, ...
  -1,  1/3,   2/3,   4/5, 13/15,  19/21,  13/14,  17/18,  43/45, ...
Numerators are listed in A176126, denominators are in A064038, and denominator - numerator = 2, 2, 1, 1,... (A014695).
  4/3, 1/3,  2/15,  1/15, 4/105,   1/42,   1/63,   1/90,  4/495, ...
  -1, -1/5, -1/15, -1/35, -1/70, -1/126, -1/210, -1/330, -1/495, ...
where the denominators of the second row are listed in A000332.
Also for those of the inverse binomial transform
  1, -1, 4/3, -1, 4/5, -2/3, 4/7, -1/2, 4/9, -2/5, 4/11, -1/3, ... ?
a(n) is the (n+1)-th term of the numerators of the first row.

			a(0) = 0/2, a(1) = 1/1, a(2) = 4/4, a(3) = 9/1.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,3,0,0,0,-3,0,0,0,1).

Cf. A000290, A000332, A016754, A026741, A060819, A061038, A109008, A139098, A168077, A176895, A181900.

[(1-(1/16)*(1+(-1)^n)*(5-(-1)^(n div 2)) )*n^2: n in [0..60]]; // Vincenzo Librandi, Jun 12 2015

seq(seq((4*i+j-1)^2/[2,1,4,1][j],j=1..4),i=0..30); # Robert Israel, May 14 2015

f[n_] := Switch[ Mod[n, 4], 0, n^2/2, 1, n^2, 2, n^2/4, 3, n^2]; Array[f, 50, 0] (* or *) Table[(4 i + j - 1)^2/{2, 1, 4, 1}[[j]], {i, 0, 12}, {j, 4}] // Flatten (* after Robert Israel *) (* or *) LinearRecurrence[{0, 0, 0, 3, 0, 0, 0, -3, 0, 0, 0, 1}, {0, 1, 1, 9, 8, 25, 9, 49, 32, 81, 25, 121}, 53] (* or *) CoefficientList[ Series[-((x (1 + x (1 + x (9 + x (8 + x (22 + x (6 + x (22 + x (8 + x (9 + x + x^2))))))))))/(-1 + x^4)^3), {x, 0, 52}], x] (* Robert G. Wilson v, May 19 2015 *)

concat(0, Vec(-x*(x^10 + x^9 + 9*x^8 + 8*x^7 + 22*x^6 + 6*x^5 + 22*x^4 + 8*x^3 + 9*x^2 + x + 1) / ((x-1)^3*(x+1)^3*(x^2+1)^3) + O(x^100))) \\ Colin Barker, May 14 2015

a(n) = n^2/(period 4: repeat 2, 1, 4, 1).
a(4n) = 8*n^2, a(2n+1) = a(4n+2) = (2*n+1)^2.
a(n+4) = a(n) + 8*A060819(n).
a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12), n>11.
4*a(n) = (period 4: repeat 2, 1, 4, 1) * A061038(n).
G.f.: -x*(x^10+x^9+9*x^8+8*x^7+22*x^6+6*x^5+22*x^4+8*x^3+9*x^2+x+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, May 14 2015
a(2n) = A181900(n), a(2n+1) = A016754(n). [Bruno Berselli, May 14 2015]
a(n) = ( 1 - (1/16)*(1+(-1)^n)*(5-(-1)^(n/2)) )*n^2. - Bruno Berselli, May 14 2015
Sum_{n>=1} 1/a(n) = 13*Pi^2/48. - Amiram Eldar, Aug 12 2022

Missing term (1521) inserted in the sequence by Colin Barker, May 14 2015
Definition uses a formula by Jean-François Alcover, Jul 01 2015
Keyword:mult added by Andrew Howroyd, Aug 06 2018

A291330 The arithmetic function v_4(n,1).

Original entry on oeis.org

0, 2, 0, 4, 4, 6, 4, 8, 8, 10, 8, 12, 12, 14, 12, 16, 16, 18, 16, 20, 20, 22, 20, 24, 24, 26, 24, 28, 28, 30, 28, 32, 32, 34, 32, 36, 36, 38, 36, 40, 40, 42, 40, 44, 44, 46, 44, 48, 48, 50, 48, 52, 52, 54, 52, 56, 56, 58, 56, 60, 60, 62, 60, 64, 64, 66, 64, 68, 68

Offset: 2

Robert Price, Aug 22 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A289435, A289436, A289437, A289438, A289439, A289440, A289441.

Cf. A109008.

seq(n-gcd(n,4), n=2..100); # Ridouane Oudra, Dec 15 2024

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[4, n, 1], {n, 2, 70}]

/* Adapted from Mathematica program */
v(g, n, h) = my(d=divisors(n)); for(k=1, #d, d[k]=floor(((d[k]-1-gcd(d[k], g))/h) + 1)*n/d[k]); vecmax(d)
a(n) = v(4, n, 1) \\ Felix Fröhlich, Aug 22 2017

a(n) = n - gcd(n,4) = n - A109008(n). - Ridouane Oudra, Dec 15 2024

Sum_{n>=5} (-1)^n/a(n) = (1 - log(2))/2. - Amiram Eldar, Jan 15 2025

cp's OEIS Frontend

A060819 a(n) = n / gcd(n,4).

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A003989 Triangle T from the array A(x, y) = gcd(x,y), for x >= 1, y >= 1, read by antidiagonals.

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A109004 Table of gcd(n, m) read by antidiagonals, n >= 0, m >= 0.

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A010121 Continued fraction for sqrt(7).

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A109012 a(n) = gcd(n,9).

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A146160 Period 4: repeat [1, 4, 1, 16].

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