A010693 Periodic sequence: Repeat 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, 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, 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, 2, 3

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) = smallest prime divisor of n!! for n >= 2. For biggest prime divisor of n!! see A139421. - Artur Jasinski, Apr 21 2008

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

Simple continued fraction of 1+sqrt(5/3) = A176020. - R. J. Mathar, Mar 08 2012

p(n) = a(n-1) is the Abelian complexity function of the Thue-Morse word A010060. - Nathan Fox, Mar 12 2013

Links

• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 466
• G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. Soc. 83(1) (2011) 79-95.
• G. Richomme, K. Saari, and L. Q. Zamboni, Abelian complexity in minimal subshifts, arXiv:0911.2914 [math.CO], 2009.
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A139421.

Cf. A026549 (partial products).

Programs

Haskell

a010693 = (+ 2) . (`mod` 2)  -- Reinhard Zumkeller, Nov 27 2012
a010693_list = cycle [2,3]  -- Reinhard Zumkeller, Mar 29 2012

Magma

[2 + (n mod 2) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2014

Maple

A010693:=n->2+(n mod 2): seq(A010693(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2014

Mathematica

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 *)
2 + Mod[Range[0, 100], 2] (* Wesley Ivan Hurt, Jul 24 2014 *)
PadRight[{},120,{2,3}] (* Harvey P. Dale, Jan 20 2023 *)

PARI

a(n)=3 - (n+1)%2 \\ Charles R Greathouse IV, May 09 2016

Formula

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

a(n) = 2 + (n mod 2) = A007395(n) + A000035(n). - Reinhard Zumkeller, Mar 23 2005

a(n) = A020639(A016767(n)) for n>0. - Reinhard Zumkeller, Jan 29 2009

From Jaume Oliver Lafont, Mar 20 2009: (Start)

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

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

a(n) = A001615(2n)/A001615(n) for n > 0. - Enrique Pérez Herrero, Jun 06 2012

a(n) = floor((n+1)*5/2) - floor((n)*5/2). - Hailey R. Olafson, Jul 23 2014

a(n) = 3 - ((n+1) mod 2). - Wesley Ivan Hurt, Jul 24 2014

E.g.f.: 2*cosh(x) + 3*sinh(x). - Stefano Spezia, Aug 04 2025

Extensions

Definition rewritten by Bruno Berselli, Sep 30 2011