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

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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

Examples

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

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).

Crossrefs

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

Complement: A047233

Programs

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

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 *)

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

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

Extensions

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