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 A035316 Sum of the square divisors of n. 46
 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 A. Dixit, B. Maji, A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937, 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 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, 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). Sequence in context: A222119 A351086 A102280 * A293718 A068316 A359945 Adjacent sequences: A035313 A035314 A035315 * A035317 A035318 A035319 KEYWORD nonn,mult,changed AUTHOR STATUS approved

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Last modified March 29 14:52 EDT 2023. Contains 361599 sequences. (Running on oeis4.)