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%I #17 Mar 07 2024 11:13:44
%S 1,1,2,1,2,1,6,1,2,1,10,1,6,3,2,1,2,1,6,1,2,5,110,1,2,3,2,3,42,1,30,1,
%T 10,1,6,1,6,3,2,1,10,1,42,5,2,55,2530,1,6,1,2,3,78,1,2,3,2,21,1218,1,
%U 30,15,2,1,6,5,330,1,110,3,210,1,6,3,2,3,30,1,78,1,2,5,410,1,2,21,14,5,110
%N a(n) is the product of those primes which divide some iterate of the Euler totient function but do not divide n itself.
%C a(n) = product of primes p such that p does not divide n but p divides phi(n)*phi(phi(n))*phi(phi(phi(n)))...
%H T. D. Noe, <a href="/A113766/b113766.txt">Table of n, a(n) for n = 1..1000</a>
%H Florian Luca and Carl Pomerance, <a href="https://www.emis.de/journals/INTEGERS/papers/a25int2005/a25int2005.pdf">Irreducible radical extensions and Euler-function chains</a>, pp. 351-362 in Combinatorial Number Theory, Landman et al., eds., de Gruyter, 2007 and in Integers, 7(2) (2007), paper A25.
%e E.g. phi(21)=12, phi(12)=4, phi(4)=2, phi(2)=1, so the only candidates are 2 and 3. But 3|21, so a(21)=2.
%e phi(43)=42, phi(42)=12, etc., so the candidates are 2, 3, 7, none of which divide 43, so a(43)=42.
%t f[n_] := Times @@ Select[First /@ FactorInteger[Times @@ FixedPointList[ EulerPhi@# &, n]], Mod[n, # ] != 0 &]; Array[f, 90] (* _Robert G. Wilson v_, Jul 08 2006 *)
%o (PARI) a(n)=my(s=n,v=[]~);while(s>1,v=concat(v,factor(s=eulerphi(s))[,1])); v=setminus(vecsort(v~,,8),factor(n)[,1]~); prod(i=1,#v,v[i]) \\ _Charles R Greathouse IV_, Feb 04 2013
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, based on an email message from R. K. Guy, Jan 19 2006
%E Corrected and extended by _Robert G. Wilson v_, Jul 08 2006