A039650 Prime reached by iterating f(x) = phi(x)+1 on n.

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, 29, 7, 31, 17, 13, 17, 13, 13, 37, 19, 13, 17, 41, 13, 43, 13, 13, 23, 47, 17, 43, 13, 13, 13, 53, 19, 41, 13, 37, 29, 59, 17, 61, 31, 37, 13, 43, 13, 67, 13, 13, 13, 71, 13, 73, 37, 41

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

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

Sequence A229487 gives the conjectured largest number that converges to prime(n). - T. D. Noe, Oct 17 2013

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

Examples

			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.
		

References

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

Links

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

Crossrefs

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

Programs

Maple

A039650 := proc(n)
    local nitr,niitr ;
    niitr := n ;
    while true do:
        nitr := 1+numtheory[phi](niitr) ;
        if nitr = niitr then
            return nitr ;
        end if;
        niitr := nitr ;
    end do:
end proc:
seq(A039650(n),n=1..40) ; # R. J. Mathar, Dec 11 2019

Mathematica

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