A171182 Period 6: repeat [0, 1, 1, 1, 0, 2].

0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 1

Offset: 1

Author

Juri-Stepan Gerasimov, Dec 04 2009, Dec 07 2009

Keywords

• nonn
• easy
• less

Comments

The number of divisors d of n of the form d=2 or 3. - Vladimir Shevelev, May 21 2010

a(n) = s(n+6), where s(k) is the number of partitions of k into distinct parts such that max(p) = 2 + min(p) for k >= 1, and (s(0)..s(6)) = (0,0,0,0,1,0,2). - Clark Kimberling, Apr 15 2014

Number of r X s integer-sided rectangles such that r < s, r + s = 2n, r | s and (s - r)/2 | s. - Wesley Ivan Hurt, Apr 24 2020

Number of positive integer solutions, (r,s,t), of the equation r^2 + t*s^2 = (n + 6)^2, where r + s = n + 6 and t < r <= s. For example, when n=6 we have the two solutions (4,8,2) and (6,6,3) since 4^2 + 2*8^2 = 12^2 and 6^2 + 3*6^2 = 12^2. - Wesley Ivan Hurt, Oct 04 2020

Links

• Vladimir Shevelev, A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n), arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, May 21 2010]
• Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Crossrefs

Cf. A178142. - Vladimir Shevelev, May 21 2010

Cf. A115357.

Cf. A001221, A059841, A065331, A079978, A089128, A169611.

Number of distinct prime factors <= p: this sequence (p=3), A178146 (p=5), A210679 (p=7).

Programs

Magma

&cat[[0, 1, 1, 1, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 19 2016

Maple

A171182:=n->(1+floor((n-3)^2/2)) mod 3: seq(A171182(n), n=1..100); # Wesley Ivan Hurt, Aug 27 2014

Mathematica

PadRight[{},110,{0,1,1,1,0,2}] (* Harvey P. Dale, Jan 28 2013 *)
LinearRecurrence[{-1, 0, 1, 1},{0, 1, 1, 1},105] (* Ray Chandler, Aug 26 2015 *)

Formula

a(n) = A115357(n-2) for n>1. - R. J. Mathar, Dec 09 2009

a(2) = 1, a(3) = 1, a(5) = 0, otherwise a(n) = a(n-2) + a(n-3) - a(n-5), where we put a(n) = 0, if n<0. - Vladimir Shevelev, May 21 2010

a(n) = floor(((n+1) mod 6)/3) + 2*floor(((n+5) mod 6)/5). - Gary Detlefs, Feb 15 2014

From Wesley Ivan Hurt, Aug 27 2014: (Start)

G.f.: (2+2*x+x^2)/(1+x-x^3-x^4).

a(n) + a(n-1) = a(n-3) + a(n-4) for n>4.

a(n) = (1 + floor((n-3)^2/2)) mod 3. (End)

a(n) = (5 + 3*cos(n*Pi) + 4*cos(2*n*Pi/3))/6. - Wesley Ivan Hurt, Jun 19 2016

From Amiram Eldar, Sep 16 2023: (Start)

Additive with a(p^e) = 1 if p <= 3, and 0 otherwise.

a(n) = A059841(n) + A079978(n).

a(n) = A001221(A089128(n)).

a(n) = A001221(A065331(n)). (End)

Extensions

Edited by Charles R Greathouse IV, Mar 23 2010