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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. 5
1, 23, 29, 47, 53, 59, 71, 83, 103, 107, 131, 149, 167, 173, 179, 191, 197, 223, 227, 239, 263, 269, 283, 293, 311, 317, 347, 359, 373, 383, 389, 419, 431, 443, 467, 479, 491, 503, 509, 557, 563, 569, 587, 599, 643, 647, 653, 659, 677, 683, 709, 719 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Prime numbers in this sequence when multiplied by 2 equal k + 2. For example, 83 * 2 = 164 + 2. - Torlach Rush, Jun 16 2018

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

K. B. Stolarski and S. Greenbaum, A Ratio Associated with phi(x) = n, The Fibonacci Quarterly, Volume 23, Number 3, August 1985, pp. 265-269.

EXAMPLE

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).

MATHEMATICA

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]

PROG

(Haskell)

import Data.List.Ordered (insertBag)

import Data.List (groupBy); import Data.Function (on)

a085713 n = a085713_list !! (n-1)

a085713_list = 1 : r yx3ss where

   r (ps:pss) | a010051' cd == 1 &&

                map (flip div cd) ps == [3, 4, 6] = cd : r pss

              | otherwise = r pss  where cd = foldl1 gcd ps

   yx3ss = filter ((== 3) . length) $

       map (map snd) $ groupBy ((==) `on` fst) $

       f [1..] a002110_list []

       where f is'@(i:is) ps'@(p:ps) yxs

              | i < p = f is ps' $ insertBag (a000010' i, i) yxs

              | otherwise = yxs' ++ f is' ps yxs''

              where (yxs', yxs'') = span ((<= a000010' i) . fst) yxs

-- Reinhard Zumkeller, Nov 25 2015

CROSSREFS

Cf. A000010, A002202, A007367, A007374, A032447, A058277, A064275.

Cf. A007614, A010051.

Sequence in context: A106988 A127834 A108111 * A102904 A108249 A232725

Adjacent sequences:  A085710 A085711 A085712 * A085714 A085715 A085716

KEYWORD

nonn

AUTHOR

Alford Arnold, Jul 19 2003

EXTENSIONS

Edited and extended by Robert G. Wilson v, Jul 19 2003

Nonprimes 343=7^3 and 361=19^2 deleted by Reinhard Zumkeller, Nov 25 2015

STATUS

approved

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Last modified April 24 16:16 EDT 2019. Contains 322430 sequences. (Running on oeis4.)