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%I #63 Feb 13 2024 20:03:27
%S 4,5,10,11,16,17,22,23,28,29,34,35,40,41,46,47,52,53,58,59,64,65,70,
%T 71,76,77,82,83,88,89,94,95,100,101,106,107,112,113,118,119,124,125,
%U 130,131,136,137,142,143,148,149
%N Numbers that are congruent to {4, 5} mod 6.
%C 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
%C 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
%D 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.
%H David Lovler, <a href="/A047257/b047257.txt">Table of n, a(n) for n = 1..1000</a>
%H The IMO Compendium, <a href="https://imomath.com/othercomp/Can/CanMO83.pdf">Problem 4</a>, 15th Canadian Mathematical Olympiad 1983.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%F a(n) = 4 + 6*floor(n/2) + n mod 2.
%F a(n) = 6*n-a(n-1)-3, with a(1)=4. - _Vincenzo Librandi_, Aug 05 2010
%F G.f.: ( x*(4+x+x^2) ) / ( (1+x)*(x-1)^2 ). - _R. J. Mathar_, Oct 08 2011
%F a(n) = 3*n - (-1)^n. - _Wesley Ivan Hurt_, Mar 20 2015
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) - log(2)/3. - _Amiram Eldar_, Dec 14 2021
%F E.g.f.: 1 + 3*x*exp(x) - exp(-x). - _David Lovler_, Aug 25 2022
%t Select[Range@ 150, 4 <= Mod[#, 6] <= 5 &] (* _Michael De Vlieger_, Mar 20 2015 *)
%t LinearRecurrence[{1,1,-1},{4,5,10},50] (* _Harvey P. Dale_, Oct 16 2017 *)
%o (Maxima) A047257(n):=4 + 6*floor(n/2) + mod(n,2)$ akelist(A047257(n),n,0,40); /* _Martin Ettl_, Oct 24 2012 */
%o (PARI) a(n) = 3*n - (-1)^n \\ _David Lovler_, Aug 25 2022
%Y Cf. A000325.
%Y Similar with: A299174 (p = 2), this sequence (p = 3), A349767 (p = 5).
%Y Cf. also A047227, A047235, A047246, A047247, A047270.
%Y Cf. A007623, A049345, A276076, A276086.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
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