A047229 Numbers that are congruent to {0, 2, 3, 4} mod 6.

0, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Appears to be the sequence of n such that n never divides 3^x-2^x for x>=0. - Benoit Cloitre, Aug 19 2002

Numbers divisible by 2 or 3. - Nick Hobson, Mar 13 2007

Numbers k such that average of squares of the first k positive integers is not integer. A089128(a(n)) > 1. For n >= 2, a(n) = complement of A007310. - Jaroslav Krizek, May 28 2010

Numbers k such that k*Fibonacci(k) is even. - Gary Detlefs, Oct 27 2011

Also numbers that have a divisor d with 2^1 <= d < 2^2 (see Ei definition p. 340 in Besicovitch article). - Michel Marcus, Oct 31 2013

Starting with 0, 2, a(n) is the smallest number greater than a(n-1) that is not relatively prime to a(n-2). - Franklin T. Adams-Watters, Dec 04 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• A. S. Besicovitch, On the density of certain sequences of integers, Mathematische Annalen, Vol. 110, No. 1 (1935), pp. 336-341; alternative link.
• Hsien-Kuei Hwang, Svante Janson and Tsung-Hsi Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, Vol. 13, No. 4 (2017), Article #47; ResearchGate link; preprint, 2016.
• Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1)

Crossrefs

Cf. A007310 (complement).

Union of A005843 and A008585.

Programs

Haskell

a047229 n = a047229_list !! (n-1)
a047229_list = filter ((`notElem` [1,5]) . (`mod` 6)) [0..]
-- Reinhard Zumkeller, Jun 30 2012

Magma

[ n : n in [0..150] | n mod 6 in [0, 2, 3, 4]] ; // Vincenzo Librandi, Jun 01 2011

Mathematica

Select[Range[0,100],MemberQ[{0,2,3,4},Mod[#,6]]&] (* Harvey P. Dale, Aug 15 2011 *)
a[ n_] := With[ {m = n - 1}, {2, 3, 4, 0}[[Mod[m, 4, 1]]] + Quotient[ m, 4] 6]; (* Michael Somos, Oct 05 2015 *)

PARI

a(n)=(n-1)\4*6+[4,0,2,3][n%4+1] \\ Charles R Greathouse IV, Oct 28 2011

Python

def A047229(n): return 3*(n-1>>1&-2)+(4,0,2,3)[n&3] # Chai Wah Wu, Nov 18 2024

Formula

a(n) = (6*(n-1) - (1+(-1)^n)*(-1)^(n*(1+(-1)^n)/4))/4; also a(n) = (6*(n-1) - (-i)^n - i^n)/4, where i is the imaginary unit. - Bruno Berselli, Nov 08 2010

G.f.: x^2*(2-x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = floor((6*n-5)/4) + floor((1/2)*cos((n+2)*Pi/2) + 1/2). - Gary Detlefs, Oct 28 2011 and corrected by Aleksey A. Solomein, Feb 08 2016

a(n) = a(n-1) + a(n-4) - a(n-5), n>4. - Gionata Neri, Apr 15 2015

a(n) = -a(2-n) for all n in Z. - Michael Somos, Oct 05 2015

a(n) = n + 2*floor((n-2)/4) + floor(f(n+2)/3), where f(n) = n mod 4. - Aleksey A. Solomein, Feb 08 2016

a(n) = (3*n - 3 - cos(n*Pi/2))/2. - Wesley Ivan Hurt, Oct 02 2017

Equals {0} union {A003586(j) * A007310(k) for j>1 and k>0}. - Flávio V. Fernandes, Jul 21 2021

Sum_{n>=2} (-1)^n/a(n) = log(3)/2 - log(2)/3. - Amiram Eldar, Dec 12 2021