A113808 - OEIS

%I #19 Jan 28 2019 08:08:59

%S 1,2,3,6,9,15,18,27,30,39,54,78,81,111,162,183,222,243,255,327,363,

%T 366,471,486,510,654,726,729,942,1458,2187,2199,3063,4359,4374,4375,

%U 4398,5571,6126,6561,8718,8750,8751,11142,13122,15723,17502,19683,31446,36759

%N Positive integers n such that S(n) divides n, where S(n) is the sum of the iterates of the Euler phi-function of n, that is, S(n) = phi(n)+phi(phi(n))+....+ 1.

%H Donovan Johnson, <a href="/A113808/b113808.txt">Table of n, a(n) for n = 1..100</a>

%H Igor E. Shparlinski, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Shparlinski/shpar43.html">On the sum of iterations of the Euler function</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.6.

%e 18 is in the sequence because phi(18)+phi(phi(18))+phi(phi(phi(18))) = 6 + 2 + 1 = 9, which divides 18.

%t s[1]=1; s[n_] := Total@NestWhileList[EulerPhi, n, #>1 &] - n; Select[Range@ 1000, Mod[#, s@#] == 0 &] (* _Giovanni Resta_, May 25 2013 *)

%o (PARI) lista(nn) = {for (n=1, nn, s = 0; m = n; until (m == 1, m = eulerphi(m); s += m;); if ((n % s == 0), print1(n, ", ")););} \\ _Michel Marcus_, May 25 2013

%Y Cf. A092693, A082897.

%K nonn

%O 1,2

%A _Jeffrey Shallit_, Jan 22 2006