A010693 - OEIS

%I #58 Mar 26 2024 18:28:07

%S 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,

%T 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,

%U 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3

%N Periodic sequence: Repeat 2,3.

%C a(n) = smallest prime divisor of n!! for n >= 2. For biggest prime divisor of n!! see A139421. - _Artur Jasinski_, Apr 21 2008

%C Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=-charpoly(A,-2). - _Milan Janjic_, Jan 27 2010

%C Simple continued fraction of 1+sqrt(5/3) = A176020. - _R. J. Mathar_, Mar 08 2012

%C p(n) = a(n-1) is the Abelian complexity function of the Thue-Morse word A010060. - _Nathan Fox_, Mar 12 2013

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=466">Encyclopedia of Combinatorial Structures 466</a>

%H G. Richomme, K. Saari, L. Q. Zamboni, <a href="http://dx.doi.org/10.1112/jlms/jdq063">Abelian complexity in minimal subshifts</a>, J. London Math. Soc. 83(1) (2011) 79-95.

%H G. Richomme, K. Saari, L. Q. Zamboni, <a href="http://arxiv.org/abs/0911.2914">Abelian complexity in minimal subshifts</a>, arXiv:0911.2914 [math.CO], 2009.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).

%F a(n) = 5/2 - ((-1)^n)/2.

%F a(n) = 2 + (n mod 2) = A007395(n) + A000035(n). - _Reinhard Zumkeller_, Mar 23 2005

%F a(n) = A020639(A016767(n)) for n>0. - _Reinhard Zumkeller_, Jan 29 2009

%F From _Jaume Oliver Lafont_, Mar 20 2009: (Start)

%F G.f.:(2+3*x)/(1-x^2).

%F Linear recurrence: a(0)=2, a(1)=3, a(n)=a(n-2) for n>=2. (End)

%F a(n) = A001615(2n)/A001615(n) for n > 0. - _Enrique Pérez Herrero_, Jun 06 2012

%F a(n) = floor((n+1)*5/2) - floor((n)*5/2). - _Hailey R. Olafson_, Jul 23 2014

%F a(n) = 3 - ((n+1) mod 2). - _Wesley Ivan Hurt_, Jul 24 2014

%p A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # _Wesley Ivan Hurt_, Jul 24 2014

%t Table[5/2 - (-1)^n/2, {n, 0, 100}] or a = {}; Do[b = First[First[FactorInteger[n!! ]]]; AppendTo[a, b], {n, 2, 1000}]; a (* _Artur Jasinski_, Apr 21 2008 *)

%t 2 + Mod[Range[0, 100], 2] (* _Wesley Ivan Hurt_, Jul 24 2014 *)

%t PadRight[{},120,{2,3}] (* _Harvey P. Dale_, Jan 20 2023 *)

%o (Haskell)

%o a010693 = (+ 2) . (`mod` 2)  -- _Reinhard Zumkeller_, Nov 27 2012

%o a010693_list = cycle [2,3]  -- _Reinhard Zumkeller_, Mar 29 2012

%o (Magma) [2 + (n mod 2) : n in [0..100]]; // _Wesley Ivan Hurt_, Jul 24 2014

%o (PARI) a(n)=3 - (n+1)%2 \\ _Charles R Greathouse IV_, May 09 2016

%Y Cf. A139421.

%Y Cf. A026549 (partial products).

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E Definition rewritten by _Bruno Berselli_, Sep 30 2011