A047255 Numbers that are congruent to {1, 2, 3, 5} mod 6.

1, 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, 26, 27, 29, 31, 32, 33, 35, 37, 38, 39, 41, 43, 44, 45, 47, 49, 50, 51, 53, 55, 56, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 80, 81, 83, 85, 86, 87, 89, 91, 92, 93, 95, 97, 98, 99, 101, 103, 104

Offset

1,2

Comments

Each element is coprime to preceding two elements. - Amarnath Murthy, Jun 12 2001

The sequence is the interleaving of A047241 with A016789. - Guenther Schrack, Feb 16 2019

Links

Guenther Schrack, Table of n, a(n) for n = 1..10012

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Formula

{k | k == 1, 2, 3, 5 (mod 6)}.

G.f.: x*(1 + x^2 + x^3) / ((1+x^2)*(1-x)^2). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, May 20 2016: (Start)

a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4), for n>4.

a(n) = (6*n - 4 + i^(1-n) + i^(n-1))/4, where i = sqrt(-1).

a(2*n) = A016789(n-1) for n>0, a(2*n-1) = A047241(n). (End)

E.g.f.: (2 + sin(x) + (3*x - 2)*exp(x))/2. - Ilya Gutkovskiy, May 21 2016

a(1-n) = - A047251(n). - Wesley Ivan Hurt, May 21 2016

From Guenther Schrack, Feb 16 2019: (Start)

a(n) = (6*n - 4 + (1 - (-1)^n)*(-1)^(n*(n-1)/2))/4.

a(n) = a(n-4) + 6, a(1)=1, a(2)=2, a(3)=3, a(4)=5, for n > 4.

a(n) = A047237(n) + 1. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 + log(2)/3 - log(3)/4. - Amiram Eldar, Dec 17 2021

a(n) = 2*n - 1 - floor(n/2) + floor(n/4) - floor((n+1)/4). - Ridouane Oudra, Feb 21 2023

Example

After 21 and 23 the next term is 25 as 24 has a common divisor with 21.

Maple

A047255:=n->(6*n-4+I^(1-n)+I^(n-1))/4: seq(A047255(n), n=1..100); # Wesley Ivan Hurt, May 20 2016

Mathematica

Select[Range[100], MemberQ[{1, 2, 3, 5}, Mod[#, 6]] &]
LinearRecurrence[{2, -2, 2, -1}, {1, 2, 3, 5}, 100] (* Harvey P. Dale, May 14 2020 *)

Prog

(Haskell)
a047255 n = a047255_list !! (n-1)
a047255_list = 1 : 2 : 3 : 5 : map (+ 6) a047255_list
-- Reinhard Zumkeller, Jan 17 2014
(Magma) [n : n in [0..100] | n mod 6 in [1, 2, 3, 5]]; // Wesley Ivan Hurt, May 21 2016
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -1, 2, -2, 2]^(n-1)*[1; 2; 3; 5])[1, 1] \\ Charles R Greathouse IV, Feb 11 2017
(Sage) a=(x*(1+x^2+x^3)/((1+x^2)*(1-x)^2)).series(x, 80).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 16 2019

Crossrefs

Cf. A007310, A016789, A047228, A047237, A047241, A047251, A047261, A047273, A062062, A062063.

Complement: A047233

Sequence in context: A186042 A039019 A359028 * A062062 A256133 A078643

Adjacent sequences: A047252 A047253 A047254 * A047256 A047257 A047258

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jun 15 2001

Status

approved