A089128 a(n) = gcd(6,n).

6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2

Offset: 0

Author

Cino Hilliard, Dec 05 2003

Keywords

• easy
• nonn
• mult

Comments

Also greatest common divisor of n^2-6 and n^2+6.

The second term of sequences of this type for n=0,1,2... form the sequence 1,2,1,2,1,... in decimal 0.1212121212... = 4/33.

Multiplicative with a(p^e) = GCD(p^e, 6). - David W. Wilson, Jun 12 2005

From Jaroslav Krizek, May 27 2010: (Start)

a(n) = denominators of averages of squares of the first n positive integers for n >= 1.

a(n) is periodic sequence with period (6, 1, 2, 3, 2, 1).

See A175485 - numerators of averages of squares of the first n positive integers.

a(n) = A175485(n) * n / A000330(n).

For n = 337 holds: a(n) = 1 and simultaneously A175485(n) is square ( = 38025 = 195^2), i.e., the number k = 195 is quadratic mean (root mean square) of the first 337 positive integers. There are other such numbers - see A084231 and A084232.

Sqrt(A175485(n) / a(n)) for n >= 1 is the harmonic mean of the first n positive integers. (End)

Links

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

Programs

Haskell

a089128 = gcd 6  -- Reinhard Zumkeller, Apr 06 2015

Mathematica

GCD[Range[0,110],6] (* or *) PadRight[{},110,{6,1,2,3,2,1}] (* Harvey P. Dale, Dec 26 2018 *)

PARI

g(n) = for(x=0,n,print1(gcd(x^2-6,x^2+6)","))

Formula

a(n) = 1 + [2|n] + 2*[3|n] + 2*[6|n] = (15 + 5*(-1)^n + 4*cos(n*Pi/3) + 12*cos(n*2*Pi/3))/6, where [x|y] is 1 if x divides y, 0 otherwise. - Mitch Harris Jun 15 2005

From R. J. Mathar, Apr 04 2011: (Start)

Dirichlet g.f.: zeta(s)*(1+1/2^s+2/3^s+2/6^s).

G.f.: (-6 - x - 2*x^2 - 3*x^3 - 2*x^4 - x^5) / ((x-1)*(1+x)*(1 + x + x^2)*(x^2 - x + 1)). (End)

a(n) = ((n-1) mod 2 + 1)*( 2*floor(((n-1) mod 3)/2) + 1). - Gary Detlefs, Dec 28 2011

Extensions

Name changed, using David W. Wilson's formula, by Franklin T. Adams-Watters, May 16 2018