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A069932 Number of k, 1<=k<=n, such that phi(k) divides n.

%I A069932 #37 Dec 07 2024 07:49:50
%S A069932 1,2,2,4,2,5,2,7,2,5,2,11,2,5,2,11,2,9,2,10,2,5,2,19,2,5,2,9,2,11,2,
%T A069932 16,2,5,2,20,2,5,2,18,2,9,2,10,2,5,2,32,2,7,2,9,2,13,2,15,2,5,2,26,2,
%U A069932 5,2,22,2,11,2,9,2,7,2,38,2,5,2,9,2,9,2,30,2,5,2,23,2,5,2,17,2,17,2,10,2,5
%N A069932 Number of k, 1<=k<=n, such that phi(k) divides n.
%C A069932 Unlike A070633, this sequence does not give the number of all integers of the form phi(k) dividing n (for some n and some m > n, phi(m) divides n).
%H A069932 Antti Karttunen, <a href="/A069932/b069932.txt">Table of n, a(n) for n = 1..65537</a>
%H A069932 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A069932 Vaclav Kotesovec, <a href="/A069932/a069932.jpg">Plot of Sum_{k=1..n} a(k)/(n*log(n)) for n = 2..65537</a> (based on b-file).
%F A069932 Asymptotically (still conjectured): Sum_{k=1..n} a(k) = C*n*log(n) + o(n*log(n)) with C = 1.5...
%F A069932 G.f.: Sum_{k>=1} 1/(1-x^phi(k)).
%F A069932 a(n) <= A070633(n). - _Antti Karttunen_, Sep 10 2018
%F A069932 a(n) = Sum_{k=1..n} (1 - ceiling(n/phi(k)) + floor(n/phi(k))). - _Wesley Ivan Hurt_, Apr 21 2023
%F A069932 a(n) = n - A378642(n). - _Paolo Xausa_, Dec 06 2024
%t A069932 a[n_] := Boole[ Divisible[n, EulerPhi[#]]] & /@ Range[n] // Total; Table[a[n], {n, 1, 94}] (* _Jean-François Alcover_, May 23 2013 *)
%o A069932 (PARI) for(n=1,150,print1(sum(i=1,n,if(n%eulerphi(i),0,1)),","))
%o A069932 (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/(1-x^eulerphi(k)),x*O(x^n)),n))
%o A069932 (PARI) A069932(n) = sum(k=1, n, !(n%eulerphi(k))); \\ _Antti Karttunen_, Sep 10 2018
%o A069932 (PARI) a(n) = sumdiv(n, d, #select(x -> x<=n, invphi(d))); \\ _Amiram Eldar_, Nov 29 2024, using _Max Alekseyev_'s invphi.gp
%Y A069932 Cf. A000010, A070633, A378642.
%Y A069932 Row lengths of A378636.
%K A069932 easy,nonn
%O A069932 1,2
%A A069932 _Benoit Cloitre_, May 05 2002