%I #27 Jun 20 2018 22:35:52
%S 1,23,29,47,53,59,71,83,103,107,131,149,167,173,179,191,197,223,227,
%T 239,263,269,283,293,311,317,347,359,373,383,389,419,431,443,467,479,
%U 491,503,509,557,563,569,587,599,643,647,653,659,677,683,709,719
%N Consider numbers k such that phi(x) = k has exactly 3 solutions and they are (3*p, 4*p, 6*p) where p is 1 or a prime. Sequence gives values of p.
%C Prime numbers in this sequence are called prime replicators of 2, by Stolarski and Greenbaum, (3, 4, 6) being the solutions of phi(x)=2. - _Michel Marcus_, Oct 20 2012
%C Prime numbers in this sequence when multiplied by 2 equal k + 2. For example, 83 * 2 = 164 + 2. - _Torlach Rush_, Jun 16 2018
%H Reinhard Zumkeller, <a href="/A085713/b085713.txt">Table of n, a(n) for n = 1..500</a>
%H K. B. Stolarski and S. Greenbaum, <a href="http://www.fq.math.ca/Scanned/23-3/stolarsky.pdf">A Ratio Associated with phi(x) = n</a>, The Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 265-269.
%e 83 is a term because the three solutions (249,332,498) to phi(x) = 164 can be written as (3*83, 4*83, 6*83).
%t t = Table[ EulerPhi[n], {n, 1, 5000}]; u = Union[ Select[t, Count[t, # ] == 3 &]]; a = {}; Do[k = 1; While[ EulerPhi[3k] != u[[n]], k++ ]; AppendTo[a, k], {n, 1, 60}]; Sort[a]
%o (Haskell)
%o import Data.List.Ordered (insertBag)
%o import Data.List (groupBy); import Data.Function (on)
%o a085713 n = a085713_list !! (n-1)
%o a085713_list = 1 : r yx3ss where
%o r (ps:pss) | a010051' cd == 1 &&
%o map (flip div cd) ps == [3, 4, 6] = cd : r pss
%o | otherwise = r pss where cd = foldl1 gcd ps
%o yx3ss = filter ((== 3) . length) $
%o map (map snd) $ groupBy ((==) `on` fst) $
%o f [1..] a002110_list []
%o where f is'@(i:is) ps'@(p:ps) yxs
%o | i < p = f is ps' $ insertBag (a000010' i, i) yxs
%o | otherwise = yxs' ++ f is' ps yxs''
%o where (yxs', yxs'') = span ((<= a000010' i) . fst) yxs
%o -- _Reinhard Zumkeller_, Nov 25 2015
%Y Cf. A000010, A002202, A007367, A007374, A032447, A058277, A064275.
%Y Cf. A007614, A010051.
%K nonn
%O 1,2
%A _Alford Arnold_, Jul 19 2003
%E Edited and extended by _Robert G. Wilson v_, Jul 19 2003
%E Nonprimes 343=7^3 and 361=19^2 deleted by _Reinhard Zumkeller_, Nov 25 2015