cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A303745 Totients t where gcd({x: phi(x)=t}) > 1.

Original entry on oeis.org

10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 184, 190, 196, 198, 204, 208, 210, 212, 220, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306
Offset: 1

Views

Author

Torlach Rush, Apr 29 2018

Keywords

Comments

If the least solution of phi(x)=t is prime then gcd({x: phi(x)=t}) is prime.
If gcd({x: phi(x)=t}) > 1 is not prime then the least solution of phi(x)=t is not prime.
For known terms if the number of solutions of x: phi(x)=t is 2 or 3 then the least solution divides the greatest solution (see A297475). - Torlach Rush, Jul 03 2018

Examples

			10 is a term because the greatest common divisor of 11 and 22, the solutions of phi(10) is 11.
2 is not a term because the greatest common divisor of 3, 4 and 6, the solutions of phi(2) is 1.

Links

Crossrefs

Cf. A000010, A002202, A032447, A297475, A007367.

Programs

  • Maple
    filter:= proc(n) local L;
L:= numtheory:-invphi(n);
L <> [] and igcd(op(L)) > 1
end proc:
select(filter, [seq(i,i=2..1000, 2)]); # Robert Israel, Jun 26 2018
  • Mathematica
    Select[Range[2, 1000, 2], GCD@@invphi[#] > 1&] (* Jean-François Alcover, Jan 31 2023, using Maxim Rytin's invphi program *)
  • PARI
    isok(n) = gcd(invphi(n)) > 1; \\ Michel Marcus, May 13 2018

Formula

gcd({x: phi(x)=t}) > 1.

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Alford Arnold, Jul 19 2003

Keywords

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

Examples

			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).

Links

Crossrefs

Cf. A000010, A002202, A007367, A007374, A032447, A058277, A064275.
Cf. A007614, A010051.

Programs

  • 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
  • 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]
  • 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

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

A071388 Numbers k such that the cardinality of the set of solutions to phi(x) = k is a prime.

Original entry on oeis.org

1, 2, 8, 10, 20, 22, 28, 30, 32, 44, 46, 48, 52, 54, 56, 58, 66, 70, 72, 78, 82, 92, 96, 102, 104, 106, 110, 116, 120, 126, 130, 132, 136, 138, 140, 148, 150, 156, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 216, 220, 222, 226, 228, 238, 240, 250, 260, 262
Offset: 1

Views

Author

Labos Elemer, May 23 2002

Keywords

Comments

All terms except 1 are even. - Robert Israel, Mar 29 2020

Examples

			InvPhi(48) = {65,104,105,112,130,140,144,156,168,180,210} has 11 terms, so 48 is a term.

Links

Crossrefs

Cf. A000010, A007366, A007367, A014197, A058277, A060668, A060670, A060674, A063512, A071386-A071389.

Programs

  • Maple
    filter:= n -> isprime(nops(numtheory:-invphi(n))):
select(filter, [$1..400]); # Robert Israel, Mar 29 2020
  • PARI
    is(k) = isprime(invphiNum(k)); \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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