cp's OEIS Frontend

A035316 Sum of the square divisors of n.

%I A035316 #61 Jul 19 2025 10:11:45
%S A035316 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,
%T A035316 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,
%U A035316 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
%N A035316 Sum of the square divisors of n.
%C A035316 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
%C A035316 Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of squares (A010052). - _Wesley Ivan Hurt_, Jun 20 2024
%H A035316 Nick Hobson, <a href="/A035316/b035316.txt">Table of n, a(n) for n = 1..1000</a>
%H A035316 A. Dixit, B. Maji, and A. Vatwani, <a href="https://arxiv.org/abs/2303.09937">Voronoi summation formula for the generalized divisor function sigma_z^k(n)</a>, arXiv:2303.09937 [math.NT], 2023, sigma_(z=2,k=2,n).
%H A035316 R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a>, arXiv:1106.4038 [math.NT] (2011), Remark 15.
%H A035316 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%F A035316 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
%F A035316 G.f.: Sum_{k>0} k^2*x^(k^2)/(1-x^(k^2)). - _Vladeta Jovovic_, Dec 13 2002
%F A035316 a(n^2) = A001157(n). - _Michel Marcus_, Jan 14 2014
%F A035316 L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017
%F A035316 Sum_{k=1..n} a(k) ~ Zeta(3/2)*n^(3/2)/3 - n/2. - _Vaclav Kotesovec_, Feb 04 2019
%F A035316 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
%F A035316 a(n) = Sum_{d|n} d * c(d), where c = A010052. - _Wesley Ivan Hurt_, Jun 20 2024
%F A035316 a(n) = Sum_{d|n} lambda(d)*d*sigma(n/d), where lambda = A008836. - _Ridouane Oudra_, Jul 18 2025
%p A035316 A035316 := proc(n)
%p A035316     local a,pe,p,e;
%p A035316     a := 1;
%p A035316     for pe in ifactors(n)[2] do
%p A035316         p := pe[1] ;
%p A035316         e := pe[2] ;
%p A035316         if type(e,'even') then
%p A035316             e := e+2 ;
%p A035316         else
%p A035316             e := e+1 ;
%p A035316         end if;
%p A035316         a := a*(p^e-1)/(p^2-1) ;
%p A035316     end do:
%p A035316     a ;
%p A035316 end proc:
%p A035316 seq(A035316(n),n=1..100) ; # _R. J. Mathar_, Oct 10 2017
%t A035316 Table[ Plus @@ Select[ Divisors@ n, IntegerQ@ Sqrt@ # &], {n, 93}] (* _Robert G. Wilson v_, Feb 19 2011 *)
%t A035316 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 *)
%o A035316 (PARI) vector(93, n, sumdiv(n, d, issquare(d)*d))
%o A035316 (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
%o A035316 (Haskell)
%o A035316 a035316 n = product $
%o A035316    zipWith (\p e -> (p ^ (e + 2 - mod e 2) - 1) `div` (p ^ 2 - 1))
%o A035316            (a027748_row n) (a124010_row n)
%o A035316 -- _Reinhard Zumkeller_, Jul 28 2014
%Y A035316 Cf. A001157, A010052, A027748, A124010, A113061 (sum cube divs).
%Y A035316 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).
%Y A035316 Cf. A008836, A000203.
%K A035316 nonn,mult
%O A035316 1,4
%A A035316 _N. J. A. Sloane_