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A085713 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.

%I A085713 #36 Jan 05 2025 19:51:37
%S A085713 1,23,29,47,53,59,71,83,103,107,131,149,167,173,179,191,197,223,227,
%T A085713 239,263,269,283,293,311,317,347,359,373,383,389,419,431,443,467,479,
%U A085713 491,503,509,557,563,569,587,599,643,647,653,659,677,683,709,719
%N A085713 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 A085713 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 A085713 Prime numbers in this sequence when multiplied by 2 equal k + 2. For example, 83 * 2 = 164 + 2. - _Torlach Rush_, Jun 16 2018
%H A085713 Amiram Eldar, <a href="/A085713/b085713.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Reinhard Zumkeller)
%H A085713 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A085713 K. B. Stolarski and S. Greenbaum, <a href="https://web.archive.org/web/2024*/https://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 A085713 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 A085713 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 A085713 (Haskell)
%o A085713 import Data.List.Ordered (insertBag)
%o A085713 import Data.List (groupBy); import Data.Function (on)
%o A085713 a085713 n = a085713_list !! (n-1)
%o A085713 a085713_list = 1 : r yx3ss where
%o A085713    r (ps:pss) | a010051' cd == 1 &&
%o A085713                 map (flip div cd) ps == [3, 4, 6] = cd : r pss
%o A085713               | otherwise = r pss  where cd = foldl1 gcd ps
%o A085713    yx3ss = filter ((== 3) . length) $
%o A085713        map (map snd) $ groupBy ((==) `on` fst) $
%o A085713        f [1..] a002110_list []
%o A085713        where f is'@(i:is) ps'@(p:ps) yxs
%o A085713               | i < p = f is ps' $ insertBag (a000010' i, i) yxs
%o A085713               | otherwise = yxs' ++ f is' ps yxs''
%o A085713               where (yxs', yxs'') = span ((<= a000010' i) . fst) yxs
%o A085713 -- _Reinhard Zumkeller_, Nov 25 2015
%o A085713 (PARI) is(p) = if(p > 1 && !isprime(p), 0, invphi(eulerphi(3*p)) == [3*p, 4*p, 6*p]); \\ _Amiram Eldar_, Nov 19 2024, using _Max Alekseyev_'s invphi.gp
%Y A085713 Cf. A000010, A002202, A007367, A007374, A032447, A058277, A064275.
%Y A085713 Cf. A007614, A010051.
%K A085713 nonn
%O A085713 1,2
%A A085713 _Alford Arnold_, Jul 19 2003
%E A085713 Edited and extended by _Robert G. Wilson v_, Jul 19 2003
%E A085713 Nonprimes 343=7^3 and 361=19^2 deleted by _Reinhard Zumkeller_, Nov 25 2015