A047257 Numbers that are congruent to {4, 5} mod 6.

4, 5, 10, 11, 16, 17, 22, 23, 28, 29, 34, 35, 40, 41, 46, 47, 52, 53, 58, 59, 64, 65, 70, 71, 76, 77, 82, 83, 88, 89, 94, 95, 100, 101, 106, 107, 112, 113, 118, 119, 124, 125, 130, 131, 136, 137, 142, 143, 148, 149

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Equivalently, numbers m such that 2^m - m is divisible by 3. Indeed, for every prime p, there are infinitely many numbers m such that 2^m - m (A000325) is divisible by p, here are numbers m corresponding to p = 3. - Bernard Schott, Dec 10 2021

Numbers k for which A276076(k) and A276086(k) are multiples of nine. For a simple proof, consider the penultimate digit in the factorial and primorial base expansions of n, A007623 and A049345. - Antti Karttunen, Feb 08 2024

References

• Michael Doob, The Canadian Mathematical Olympiad & L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society & Société Mathématique du Canada, Problem 4, 1983, page 158, 1993.

Links

• David Lovler, Table of n, a(n) for n = 1..1000
• The IMO Compendium, Problem 4, 15th Canadian Mathematical Olympiad 1983.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
• Index to sequences related to Olympiads.

Crossrefs

Cf. A000325.

Similar with: A299174 (p = 2), this sequence (p = 3), A349767 (p = 5).

Cf. also A047227, A047235, A047246, A047247, A047270.

Cf. A007623, A049345, A276076, A276086.

Programs

Mathematica

Select[Range@ 150, 4 <= Mod[#, 6] <= 5 &] (* Michael De Vlieger, Mar 20 2015 *)
LinearRecurrence[{1,1,-1},{4,5,10},50] (* Harvey P. Dale, Oct 16 2017 *)

Maxima

A047257(n):=4 + 6*floor(n/2) + mod(n,2)$ akelist(A047257(n),n,0,40); /* Martin Ettl, Oct 24 2012 */

PARI

a(n) = 3*n - (-1)^n \\ David Lovler, Aug 25 2022

Formula

a(n) = 4 + 6*floor(n/2) + n mod 2.

a(n) = 6*n-a(n-1)-3, with a(1)=4. - Vincenzo Librandi, Aug 05 2010

G.f.: ( x*(4+x+x^2) ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011

a(n) = 3*n - (-1)^n. - Wesley Ivan Hurt, Mar 20 2015

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) - log(2)/3. - Amiram Eldar, Dec 14 2021

E.g.f.: 1 + 3*x*exp(x) - exp(-x). - David Lovler, Aug 25 2022