A069932 - OEIS

%I #23 Apr 21 2023 21:27:14

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

%C 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 Antti Karttunen, <a href="/A069932/b069932.txt">Table of n, a(n) for n = 1..65537</a>

%H 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 Asymptotically (still conjectured): sum(k=1, n, a(k)) = C*n*log(n) + o(n*log(n)) with C=1.5...

%F G.f.: Sum_{k>=1} 1/(1-x^phi(k)).

%F a(n) <= A070633(n). - _Antti Karttunen_, Sep 10 2018

%F a(n) = Sum_{k=1..n} (1 - ceiling(n/phi(k)) + floor(n/phi(k))). - _Wesley Ivan Hurt_, Apr 21 2023

%t a[n_] := Boole[ Divisible[n, EulerPhi[#]]] & /@ Range[n] // Total; Table[a[n], {n, 1, 94}] (* _Jean-François Alcover_, May 23 2013 *)

%o (PARI) for(n=1,150,print1(sum(i=1,n,if(n%eulerphi(i),0,1)),","))

%o (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/(1-x^eulerphi(k)),x*O(x^n)),n))

%o (PARI) A069932(n) = sum(k=1, n, !(n%eulerphi(k))); \\ _Antti Karttunen_, Sep 10 2018

%Y Cf. A000010, A070633.

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, May 05 2002