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A092694 Product of iterated phi(n).

%I A092694 #14 Nov 15 2021 01:27:48
%S A092694 1,1,2,2,8,2,12,8,12,8,80,8,96,12,64,64,1024,12,216,64,96,80,1760,64,
%T A092694 1280,96,216,96,2688,64,1920,1024,1280,1024,1536,96,3456,216,1536,
%U A092694 1024,40960,96,4032,1280,1536,1760,80960,1024,4032,1280,32768,1536,79872,216
%N A092694 Product of iterated phi(n).
%C A092694 A logarithmic plot of this sequence shows an unusual banded structure.
%H A092694 T. D. Noe, <a href="/A092694/b092694.txt">Table of n, a(n) for n=1..10000</a>
%H A092694 T. D. Noe, <a href="/A092694/a092694.gif">Plot of A092694</a>
%F A092694 a(1) = 1, a(n) = phi(n) * a(phi(n))
%e A092694 a(100) = 40960 because the iterations of phi (40, 16, 8, 4, 2, 1) have a product of 40960.
%t A092694 nMax=100; a=Table[1, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e*a[[e]], {n, 2, nMax}]; a
%o A092694 (Haskell)
%o A092694 a092694 n = snd $ until ((== 1) . fst) f (a000010 n, 1) where
%o A092694    f (x, p) = (a000010 x, p * x)
%o A092694 -- _Reinhard Zumkeller_, Jan 30 2014
%o A092694 (Python)
%o A092694 from sympy import totient
%o A092694 from math import prod
%o A092694 def f(n):
%o A092694     m = n
%o A092694     while m > 1:
%o A092694         m = totient(m)
%o A092694         yield m
%o A092694 def A092694(n): return prod(f(n)) # _Chai Wah Wu_, Nov 14 2021
%Y A092694 Cf. A003434 (iterations of phi(n) needed to reach 1), A092693 (iterated phi sum).
%Y A092694 Cf. A000010.
%K A092694 nonn,look
%O A092694 1,3
%A A092694 _T. D. Noe_, Mar 04 2004