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A035316 Sum of square divisors of n. 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

Nick Hobson, Table of n, a(n) for n = 1..1000

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.

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

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 *)

PROG

(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

(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

CROSSREFS

Cf. A001157, A010052, A046951, A027748, A124010, A113061 (sum cube divs).

Sequence in context: A334987 A222119 A102280 * A293718 A068316 A284252

Adjacent sequences:  A035313 A035314 A035315 * A035317 A035318 A035319

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 28 02:22 EST 2020. Contains 338699 sequences. (Running on oeis4.)