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
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
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