A069290 Sum of the square roots of the 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

Author

Reinhard Zumkeller, Mar 14 2002

Keywords

• nonn
• easy
• mult

Comments

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

Examples

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

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).

Crossrefs

Cf. A000005 (tau), A000188, A000203 (sigma), A001620, A005117, A035316, A037213, A046951.

Programs

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 *)

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

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

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

a(n) = Sum_{d|n} d^(1/2)*(1-(-1)^tau(d))/2, [See Mathar comment]. - Wesley Ivan Hurt, Jul 09 2025

Extensions

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