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
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.
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
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
KEYWORD
nonn,mult
AUTHOR
STATUS
approved