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A047229
Numbers that are congruent to {0, 2, 3, 4} mod 6.
15
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
1,2
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
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.
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
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 *)
PROG
(Magma) [ n : n in [0..150] | n mod 6 in [0, 2, 3, 4]] ; // Vincenzo Librandi, Jun 01 2011
(PARI) a(n)=(n-1)\4*6+[4, 0, 2, 3][n%4+1] \\ Charles R Greathouse IV, Oct 28 2011
(Haskell)
a047229 n = a047229_list !! (n-1)
a047229_list = filter ((`notElem` [1, 5]) . (`mod` 6)) [0..]
-- Reinhard Zumkeller, Jun 30 2012
(Python)
def A047229(n): return 3*(n-1>>1&-2)+(4, 0, 2, 3)[n&3] # Chai Wah Wu, Nov 18 2024
CROSSREFS
Cf. A007310 (complement).
Union of A005843 and A008585.
Sequence in context: A353684 A377236 A377182 * A094229 A067290 A106577
KEYWORD
nonn,easy
STATUS
approved