

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
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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. 265269.


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 !! (n1)
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 A338324
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



