

A047229


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


13



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
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OFFSET

1,2


COMMENTS

Appears to be the sequence of n such that n never divides 3^x2^x for x>=0.  Benoit Cloitre, Aug 19 2002
Numbers divisible by 2 or 3.  Nick Hobson (nickh(AT)qbyte.org), 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 n such that n*Fibonacci(n) 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(n1) that is not relatively prime to a(n2).  Franklin T. AdamsWatters, Dec 04 2014


REFERENCES

HsienKuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wpcontent/files/2016/12/aathhrr1.pdf. Also Exact and Asymptotic Solutions of a DivideandConquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585


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, 1935, Volume 110, Issue 1, pp 336341.
A. S. Besicovitch, On the density of certain sequences of integers, Mathematische Annalen, 1935, Volume 110, Issue 1, pp 336341.
Index entries for linear recurrences with constant coefficients, signature (2,2,2,1)


FORMULA

a(n) = (1/8)*{11*(n mod 4)+[(n+1) mod 4]+[(n+2) mod 4][(n+3) mod 4]} + 6*A002265(n).  Paolo P. Lava, Nov 05 2007
a(n) = (6*(n1)(1+(1)^n)*(1)^(n*(1+(1)^n)/4))/4; also a(n) = (6*(n1)(i)^ni^n)/4, where i is the imaginary unit.  Bruno Berselli, Nov 08 2010
G.f.: x^2*(2x+2*x^2) / ( (x^2+1)*(x1)^2 ).  R. J. Mathar, Oct 08 2011
a(n) = floor((6*n5)/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(n1) + a(n4)  a(n5), 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((n2)/4) + floor(f(n+2)/3), where f(n) = n mod 4.  Aleksey A. Solomein, Feb 08 2016
a(n) = (3*n3cos(n*Pi/2))/2.  Wesley Ivan Hurt, Oct 02 2017


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)=(n1)\4*6+[4, 0, 2, 3][n%4+1] \\ Charles R Greathouse IV, Oct 28 2011
(Haskell)
a047229 n = a047229_list !! (n1)
a047229_list = filter ((`notElem` [1, 5]) . (`mod` 6)) [0..]
 Reinhard Zumkeller, Jun 30 2012


CROSSREFS

Cf. A007310 (complement).
Union of A005843 and A008585.
Sequence in context: A173905 A171581 A063450 * A094229 A067290 A106577
Adjacent sequences: A047226 A047227 A047228 * A047230 A047231 A047232


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



