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.

1, 2, 4, 6, 9, 12, 16, 19, 23, 27, 31, 35, 40, 44, 49, 54, 59, 64, 69, 74, 79, 84, 90, 96, 102, 108, 114, 120, 125, 131, 136, 142, 149, 155, 161, 167, 173, 178, 185, 191, 198, 204, 210, 217, 223, 229, 235, 241, 248, 254, 261, 268, 275, 282, 290, 297, 304, 310

Offset: 1

Author

Labos Elemer, Feb 28 2000

Keywords

• nonn

Comments

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).

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A002110, A003434.

Programs

Mathematica

Array[-2 + Length@ FixedPointList[EulerPhi, Product[Prime@ i, {i, #}]] &, 58] (* Michael De Vlieger, Nov 20 2017 *)

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

PARI

A003434(n)=my(s);while(n>1,n=eulerphi(n);s++);s
first(n)=my(s=1); vector(n,k,s+=A003434(prime(k))-1) \\ Charles R Greathouse IV, Jan 06 2016

Formula

a(n) is the smallest number such that Nest[EulerPhi, A002110, a(n)]=1