A035316 Sum of the square divisors of n.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Comments

The Dirichlet generating function is zeta(s)*zeta(2s-2). The sequence is the Dirichlet convolution of A000012 with the sequence defined by n*A010052(n). - R. J. Mathar, Feb 18 2011

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Links

• Nick Hobson, Table of n, a(n) for n = 1..1000
• A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma_(z=2,k=2,n).
• R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.
• Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Crossrefs

Cf. A001157, A010052, A027748, A124010, A113061 (sum cube divs).

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), this sequence (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A008836, A000203.

Programs

Haskell

a035316 n = product $
   zipWith (\p e -> (p ^ (e + 2 - mod e 2) - 1) `div` (p ^ 2 - 1))
           (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 28 2014

Maple

A035316 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        if type(e,'even') then
            e := e+2 ;
        else
            e := e+1 ;
        end if;
        a := a*(p^e-1)/(p^2-1) ;
    end do:
    a ;
end proc:
seq(A035316(n),n=1..100) ; # R. J. Mathar, Oct 10 2017

Mathematica

Table[ Plus @@ Select[ Divisors@ n, IntegerQ@ Sqrt@ # &], {n, 93}] (* Robert G. Wilson v, Feb 19 2011 *)
f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

PARI

vector(93, n, sumdiv(n, d, issquare(d)*d))

PARI

a(n)=my(f=factor(n));prod(i=1,#f[,1],(f[i,1]^(f[i,2]+2-f[i,2]%2)-1)/(f[i,1]^2-1)) \\ Charles R Greathouse IV, May 20 2013

Formula

Multiplicative with a(p^e)=(p^(e+2)-1)/(p^2-1) for even e and a(p^e)=(p^(e+1)-1)/(p^2-1) for odd e. - Vladeta Jovovic, Dec 05 2001

G.f.: Sum_{k>0} k^2*x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic, Dec 13 2002

a(n^2) = A001157(n). - Michel Marcus, Jan 14 2014

L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017

Sum_{k=1..n} a(k) ~ Zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Feb 04 2019

a(n) = Sum_{k=1..n} k * (floor(sqrt(k)) - floor(sqrt(k-1))) * (1 - ceiling(n/k) + floor(n/k)). - Wesley Ivan Hurt, Jun 13 2021

a(n) = Sum_{d|n} d * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024

a(n) = Sum_{d|n} lambda(d)*d*sigma(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025