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A053096 When the Euler phi function is iterated with initial value A002110(n) = primorial, a(n) = number of iterations required to reach the fixed number = 1.

%I A053096 #14 Nov 20 2017 23:25:14
%S A053096 1,2,4,6,9,12,16,19,23,27,31,35,40,44,49,54,59,64,69,74,79,84,90,96,
%T A053096 102,108,114,120,125,131,136,142,149,155,161,167,173,178,185,191,198,
%U A053096 204,210,217,223,229,235,241,248,254,261,268,275,282,290,297,304,310
%N A053096 When the Euler phi function is iterated with initial value A002110(n) = primorial, a(n) = number of iterations required to reach the fixed number = 1.
%C A053096 Analogous to A053025, A053034, A053044. For comparison: iteration of, e.g., A000005 to primorial i.v. is trivially computable: q(n)=A002110(n), d(q(n)) = 2^n, d(d(q(n))) = n+1 and so A036450(A002110(n)) = A000005(n+1).
%H A053096 Charles R Greathouse IV, <a href="/A053096/b053096.txt">Table of n, a(n) for n = 1..10000</a>
%F A053096 a(n) is the smallest number such that Nest[EulerPhi, A002110, a(n)]=1
%e A053096 n=7, A002110(7)=510510; the corresponding iteration chain is {510510, 92160, 24576, 8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}. Its length is 17, so the required number of iterations is a(7)=16.
%t A053096 Array[-2 + Length@ FixedPointList[EulerPhi, Product[Prime@ i, {i, #}]] &, 58] (* _Michael De Vlieger_, Nov 20 2017 *)
%o A053096 (PARI) a(n)=my(t=prod(i=1,n,prime(i)-1),s=1); while(t>1, t=eulerphi(t); s++); s \\ _Charles R Greathouse IV_, Jan 06 2016
%o A053096 (PARI) A003434(n)=my(s);while(n>1,n=eulerphi(n);s++);s
%o A053096 first(n)=my(s=1); vector(n,k,s+=A003434(prime(k))-1) \\ _Charles R Greathouse IV_, Jan 06 2016
%Y A053096 Cf. A000010, A002110, A003434.
%K A053096 nonn
%O A053096 1,2
%A A053096 _Labos Elemer_, Feb 28 2000