A092694 Product of iterated phi(n).

1, 1, 2, 2, 8, 2, 12, 8, 12, 8, 80, 8, 96, 12, 64, 64, 1024, 12, 216, 64, 96, 80, 1760, 64, 1280, 96, 216, 96, 2688, 64, 1920, 1024, 1280, 1024, 1536, 96, 3456, 216, 1536, 1024, 40960, 96, 4032, 1280, 1536, 1760, 80960, 1024, 4032, 1280, 32768, 1536, 79872, 216

Offset: 1

Author

T. D. Noe, Mar 04 2004

Keywords

• nonn
• look

Comments

A logarithmic plot of this sequence shows an unusual banded structure.

Examples

			a(100) = 40960 because the iterations of phi (40, 16, 8, 4, 2, 1) have a product of 40960.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• T. D. Noe, Plot of A092694

Crossrefs

Cf. A003434 (iterations of phi(n) needed to reach 1), A092693 (iterated phi sum).

Cf. A000010.

Programs

Haskell

a092694 n = snd $ until ((== 1) . fst) f (a000010 n, 1) where
   f (x, p) = (a000010 x, p * x)
-- Reinhard Zumkeller, Jan 30 2014

Mathematica

nMax=100; a=Table[1, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e*a[[e]], {n, 2, nMax}]; a

Python

from sympy import totient
from math import prod
def f(n):
    m = n
    while m > 1:
        m = totient(m)
        yield m
def A092694(n): return prod(f(n)) # Chai Wah Wu, Nov 14 2021

Formula

a(1) = 1, a(n) = phi(n) * a(phi(n))