A178143 Sum of squares d^2 over the divisors d=2 and/or d=3 of n.

0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4, 0, 13, 0, 4, 9, 4

Offset

1,2

Comments

Period 6: repeat [0, 4, 9, 4, 0, 13]. - Wesley Ivan Hurt, Jul 05 2016

Links

Table of n, a(n) for n=1..82.

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.

Index entries for linear recurrences with constant coefficients, signature (-1,0,1,1).

Formula

a(n) = Sum_{d|n, d=2 and/or d=3} d^2.

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

G.f.: x*(4+13*x+13*x^2) / ( (1-x)*(1+x)*(1+x+x^2) ).

a(n+6) = a(n).

a(n) = A010675(n) + A021115(n). [R. J. Mathar, May 28 2010]

a(n) = 4 * (1 + floor(n/2) - ceiling(n/2)) + 9 * (1 + floor(n/3) - ceiling(n/3)). - Wesley Ivan Hurt, May 20 2013

a(n) = 5 + 2*cos(n*Pi) + 6*cos(2*n*Pi/3). - Wesley Ivan Hurt, Jul 05 2016

Example

a(1)=0, a(2)=2^2=4 since 2|2, a(3)=3^2=9 since 3|3, a(4)=2^2=4 since 2|4.

Maple

seq(op([0, 4, 9, 4, 0, 13]), n=1..30); # Wesley Ivan Hurt, Jul 05 2016

Mathematica

PadRight[{}, 100, {0, 4, 9, 4, 0, 13}] (* Wesley Ivan Hurt, Jul 05 2016 *)

Prog

(PARI) a(n)=[13, 0, 4, 9, 4, 0][n%6+1] \\ Charles R Greathouse IV, May 21 2013
(Magma) &cat [[0, 4, 9, 4, 0, 13]^^20]; // Wesley Ivan Hurt, Jul 05 2016

Crossrefs

Cf. A010675, A021115, A098002, A178142.

Sequence in context: A285323 A365325 A321219 * A070435 A070516 A143298

Adjacent sequences: A178140 A178141 A178142 * A178144 A178145 A178146

Keyword

nonn,easy,less

Author

Vladimir Shevelev, May 21 2010

Extensions

Replaced recurrence by a shorter one; added keyword:less - R. J. Mathar, May 28 2010

Status

approved