A069290 Sum of square roots of square divisors of n.

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 7, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 7, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 7, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 15, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 7, 13, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 7, 1, 8, 4, 18, 1, 1

Offset

1,4

Comments

a(m)=1 iff m is squarefree (A005117).

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=1,k=2,n).

Formula

Multiplicative with a(p^e) = (p^(floor(e/2)+1)-1)/(p-1). - Vladeta Jovovic, Apr 23 2002

G.f.: Sum_{k>=1} k*x^k^2/(1-x^k^2). - Ralf Stephan, Apr 21 2003

Dirichlet g.f.: zeta(2s-1)*zeta(s). Inverse Mobius transform of A037213. - R. J. Mathar, Oct 31 2011

Sum_{k=1..n} a(k) ~ n/2 * (log(n) - 1 + 3*gamma), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 31 2019

a(n) = Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)) * k. - Wesley Ivan Hurt, Jan 28 2021

a(n) = A000203(A000188(n)). - Amiram Eldar, Sep 01 2023

Example

Square divisors for n=48: {1, 2^2, 4^2}, so a(48) = 1+2+4 = 7.

Mathematica

nn = 102; f[list_, i_] := list[[i]]; a =Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 1, nn}]; b =Table[1, {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 21 2015 *)
f[p_, e_] := (p^(Floor[e/2] + 1) - 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

Prog

(PARI) vector(102, n, sumdiv(n, d, issquare(d)*sqrtint(d)))
(PARI) a(n)={my(s=0); fordiv(n, d, if(issquare(d), s+=sqrtint(d))); s; } \\ Joerg Arndt, Feb 22 2015
(Python 3.8+)
from math import prod
from sympy import factorint
def A069290(n): return prod((p**(q//2+1)-1)//(p-1) for p, q in factorint(n).items()) # Chai Wah Wu, Jun 14 2021

Crossrefs

Cf. A000188, A000203, A001620, A005117, A035316, A000188, A046951.

Sequence in context: A295295 A365336 A365404 * A365334 A348660 A076476

Adjacent sequences: A069287 A069288 A069289 * A069291 A069292 A069293

Keyword

nonn,easy,mult

Author

Reinhard Zumkeller, Mar 14 2002

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jul 01 2002

Status

approved