A010875 a(n) = n mod 6.

0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Period 6: repeat [0, 1, 2, 3, 4, 5].

The rightmost digit in the base-6 representation of n. - Hieronymus Fischer, Jun 11 2007

[a(n) * a(m)] mod 6 == a(n*m mod 6) == a(n*m). - Jon Perry, Nov 11 2014

If n > 3 and (a(n) is in {0,2,3,4}), then n is not prime. - Jean-Marc Rebert, Jul 22 2015, corrected by M. F. Hasler, Jul 24 2015

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65538
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Crossrefs

Partial sums: A130484. Other related sequences A130481, A130482, A130483, A130485.

Cf. also A079979, A097325, A122841.

Programs

Magma

[n mod 6: n in [0..100]]; // Wesley Ivan Hurt, Jul 06 2014

Maple

A010875:=n->n mod 6; seq(A010875(n), n=0..100); # Wesley Ivan Hurt, Jul 06 2014

Mathematica

Mod[Range[0, 100], 6] (* Wesley Ivan Hurt, Jul 06 2014 *)

PARI

a(n)=n%6 \\ Charles R Greathouse IV, Dec 05 2011

Sage

[power_mod(n,3,6 )for n in range(0, 81)] # Zerinvary Lajos, Oct 29 2009

Scheme

(define (A010875 n) (modulo n 6)) ;; Antti Karttunen, Dec 22 2017

Formula

Complex representation: a(n) = (1/6) * (1 - r^n) * Sum_{k = 1..6} k * Product_{1 <= m < 6, m <> k} (1-r^(n-m)), where r = exp((Pi/3)*i) = (1 + sqrt(3)*i)/2 and i = sqrt(-1).

Trigonometric representation: a(n) = (16/3)^2 * (sin(n*Pi/6))^2 * Sum_{k = 1..6} k * Product_{1 <= m < 6, m<>k} (sin((n-m)*Pi/6))^2.

G.f.: g(x) = (Sum_{k = 1..6} k*x^k)/(1-x^6).

Also: g(x) = x*(5*x^6 - 6*x^5 + 1)/((1 - x^6)*(1 - x)^2). - Hieronymus Fischer, May 31 2007

a(n) = (n mod 2) + 2(floor(n/2) mod 3) = A000035(n) +2*A010872(A004526(n));

a(n) = (n mod 3) + 3(floor(n/3) mod 2) = A010872(n) +3*A000035(A002264(n)). - Hieronymus Fischer, Jun 11 2007

a(n) = 2.5 - 0.5*(-1)^n - cos(Pi*n/3) - 3^0.5*sin(Pi*n/3) -cos(2*Pi*n/3) - 3^0.5/3*sin(2*Pi*n/3). - Richard Choulet, Dec 11 2008

a(n) = n^3 mod 6. - Zerinvary Lajos, Oct 29 2009

a(n) = floor(12345/999999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 03 2013

a(n) = floor(373/9331*6^(n+1)) mod 6. - Hieronymus Fischer, Jan 04 2013

a(n) = 5/2 - (-1)^n/2 - 2*0^((-1)^(n/6 - 1/12 + (-1)^n/12) - (-1)^(n/2 - 1/4 +(-1)^n/4)) + 2*0^((-1)^(n/6 + 1/4 + (-1)^n/12) + (-1)^(n/2 - 1/4 + (-1)^n/4)). - Wesley Ivan Hurt, Jun 23 2015

E.g.f.: -sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2) - 2*cosh(x/2)*cos(sqrt(3)*x/2). - Robert Israel, Jul 22 2015

Extensions

Formulas 1 to 6 re-edited for better readability by Hieronymus Fischer, Dec 05 2011

More terms from Antti Karttunen, Dec 22 2017