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 A069208 #52 Jul 01 2025 10:05:27
%S A069208 1,2,3,5,5,6,7,11,10,10,11,15,13,14,15,23,17,20,19,25,21,22,23,33,26,
%T A069208 26,31,35,29,30,31,47,33,34,35,50,37,38,39,55,41,42,43,55,50,46,47,69,
%U A069208 50,52,51,65,53,62,55,77,57,58,59,75,61,62,70,95,65,66,67,85,69,70,71
%N A069208 a(n) = Sum_{ d divides n } phi(n)/phi(d).
%C A069208 a(n) = n iff n is squarefree number (cf. A005117).
%C A069208 Conjecture: Let (f(n)), n > 0, be a multiplicative sequence. Then holds:
%C A069208 (1) p(f; n) = Sum_{d powerful number (A001694) dividing n} f(d) is multiplicative;
%C A069208 (2) p(f; n) equals inverse Moebius transform of A112526(n) * f(n). - _Werner Schulte_, Jan 23 2025
%C A069208 a(n) is also the number of conjugacy classes of the holomorph of the cyclic group of order n. Corollary: Let Rn be the dihedral quandle of order n. Then a(n) is the number of isomorphism classes of virtual quandles whose underlying quandle is isomorphic to Rn. - _Luc Ta_, Jun 16 2025
%H A069208 Ivan Neretin, <a href="/A069208/b069208.txt">Table of n, a(n) for n = 1..10000</a>
%H A069208 Lực Ta, <a href="https://arxiv.org/abs/2506.16536">Enumeration of virtual quandles up to isomorphism</a>, arXiv:2506.16536 [math.GT], 2025. See p. 2.
%F A069208 Multiplicative with a(p^e) = (p^(e+1)-p^e+p^(e-1)-1)/(p-1).
%F A069208 a(n) = phi(n) * Sum_{k=1..n} 1/phi(n / gcd(n, k))^2. - _Daniel Suteu_, Nov 04 2018
%F A069208 a(n) = Sum_{k=1..n, gcd(n,k) = 1} tau(gcd(n,k-1)). - _Ilya Gutkovskiy_, Sep 24 2021
%F A069208 From _Werner Schulte_, Feb 27 2022: (Start)
%F A069208 Dirichlet convolution of A005361 and A000010.
%F A069208 Dirichlet convolution of A112526 and A000027.
%F A069208 Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-1) * zeta(2*s) * zeta(3*s) / zeta(6*s). (End)
%F A069208 Sum_{k=1..n} a(k) ~ c * n^2, where c = 15015/(2764*Pi^2) = 0.550411... . - _Amiram Eldar_, Oct 22 2022
%F A069208 a(n) = Sum_{d powerful number (A001694) dividing n} n / d. - _Werner Schulte_, Jan 23 2025 (see Golomb link at A001694)
%t A069208 Table[EulerPhi[n]*Total[1/EulerPhi@Divisors@n], {n, 71}] (* _Ivan Neretin_, Sep 20 2017 *)
%t A069208 f[p_, e_] := (p^(e + 1) - p^e + p^(e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 14 2022 *)
%o A069208 (PARI) a(n) = sumdiv(n, d, eulerphi(n)/eulerphi(d)) \\ _Michel Marcus_, Jun 17 2013
%o A069208 (PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, (f[k,1]^(f[k,2]-1) + (f[k,1]-1)*f[k,1]^f[k,2]-1) / (f[k,1]-1)); \\ _Daniel Suteu_, Nov 04 2018
%Y A069208 Cf. A000010, A000027, A005117, A005361, A069170, A112526.
%K A069208 nonn,mult
%O A069208 1,2
%A A069208 _Vladeta Jovovic_, Apr 10 2002