This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069290 #60 Jul 09 2025 21:34:49
%S A069290 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,
%T A069290 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,
%U A069290 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
%N A069290 Sum of the square roots of the square divisors of n.
%C A069290 a(m)=1 iff m is squarefree (A005117).
%H A069290 Nick Hobson, <a href="/A069290/b069290.txt">Table of n, a(n) for n = 1..1000</a>
%H A069290 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=1,k=2,n).
%F A069290 Multiplicative with a(p^e) = (p^(floor(e/2)+1)-1)/(p-1). - _Vladeta Jovovic_, Apr 23 2002
%F A069290 G.f.: Sum_{k>=1} k*x^k^2/(1-x^k^2). - _Ralf Stephan_, Apr 21 2003
%F A069290 Dirichlet g.f.: zeta(2s-1)*zeta(s). Inverse Mobius transform of A037213. - _R. J. Mathar_, Oct 31 2011
%F A069290 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
%F A069290 a(n) = Sum_{k=1..n} (1 - ceiling(n/k^2) + floor(n/k^2)) * k. - _Wesley Ivan Hurt_, Jan 28 2021
%F A069290 a(n) = A000203(A000188(n)). - _Amiram Eldar_, Sep 01 2023
%F A069290 a(n) = Sum_{d|n} d^(1/2)*(1-(-1)^tau(d))/2, [See Mathar comment]. - _Wesley Ivan Hurt_, Jul 09 2025
%e A069290 Square divisors for n=48: {1, 2^2, 4^2}, so a(48) = 1+2+4 = 7.
%t A069290 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 *)
%t A069290 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 *)
%o A069290 (PARI) vector(102, n, sumdiv(n, d, issquare(d)*sqrtint(d)))
%o A069290 (PARI) a(n)={my(s=0);fordiv(n,d,if(issquare(d),s+=sqrtint(d)));s;} \\ _Joerg Arndt_, Feb 22 2015
%o A069290 (Python)
%o A069290 from math import prod
%o A069290 from sympy import factorint
%o A069290 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
%Y A069290 Cf. A000005 (tau), A000188, A000203 (sigma), A001620, A005117, A035316, A037213, A046951.
%K A069290 nonn,easy,mult
%O A069290 1,4
%A A069290 _Reinhard Zumkeller_, Mar 14 2002
%E A069290 More terms from Larry Reeves (larryr(AT)acm.org), Jul 01 2002