%I #19 Dec 11 2019 08:19:33

%S 2,2,3,3,5,3,7,5,7,5,11,5,13,7,7,7,17,7,19,7,13,11,23,7,13,13,19,13,

%T 29,7,31,17,13,17,13,13,37,19,13,17,41,13,43,13,13,23,47,17,43,13,13,

%U 13,53,19,41,13,37,29,59,17,61,31,37,13,43,13,67,13,13,13,71,13,73,37,41

%N Prime reached by iterating f(x) = phi(x)+1 on n.

%C Or, a(n) = lim_k {s(k,n)} where s(k,n) is defined inductively on k by: s(1,n) = n; s(k+1,n) = 1 + phi(s(k,n)). - _Joseph L. Pe_, Apr 30 2002

%C Sequence A229487 gives the conjectured largest number that converges to prime(n). - _T. D. Noe_, Oct 17 2013

%C For n>1, phi(n) <= n-1, with equality iff n is prime. So the trajectory decreases until it hits a prime. So a(n) always exists. - _N. J. A. Sloane_, Sep 22 2017

%D Alexander S. Karpenko, Lukasiewicz Logics and Prime Numbers, Luniver Press, Beckington, 2006, p. 51.

%H T. D. Noe, <a href="/A039650/b039650.txt">Table of n, a(n) for n = 1..10000</a>

%e s(24,1) = 24, s(24,2) = 1 + phi(24) = 1 + 8 = 9, s(24,3) = 1 + phi(9) = 1 + 6 = 7, s(24,4) = 1 + phi(7) = 1 + 6 = 7,.... Therefore a(24) = lim_k {s(24,k)} = 7.

%p A039650 := proc(n)

%p local nitr,niitr ;

%p niitr := n ;

%p while true do:

%p nitr := 1+numtheory[phi](niitr) ;

%p if nitr = niitr then

%p return nitr ;

%p end if;

%p niitr := nitr ;

%p end do:

%p end proc:

%p seq(A039650(n),n=1..40) ; # _R. J. Mathar_, Dec 11 2019

%t f[n_] := FixedPoint[1 + EulerPhi[ # ] &, n]; Table[ f[n], {n, 1, 75}]

%Y Cf. A039649, A039650, A039651, A039652, A039653, A039654, A039655, A039656, A229487.

%K nonn

%O 1,1

%A _David W. Wilson_