A007614 -id:A007614 - cp's OEIS frontend

A002202 Values taken by totient function phi(m) (A000010).

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176

Offset: 1

N. J. A. Sloane

These are the numbers n such that for some m the multiplicative group mod m has order n.
Maier & Pomerance show that there are about x * exp(c (log log log x)^2)/log x members of this sequence up to x, with c = 0.81781465... (A234614); see the paper for details on making this precise. - Charles R Greathouse IV, Dec 28 2013
A264739(a(n)) = 1; a(n) occurs A058277(n) times in A007614. - Reinhard Zumkeller, Nov 26 2015
There are no odd numbers > 2 in the sequence and the even numbers that are not in the sequence are in A005277. - Bernard Schott, May 13 2020

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
André Contiero, and Davi Lima, 2-Adic Stratification of Totients, arXiv:2005.05475 [math.NT], 2020.
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phi-function, Acta Arithmetica 49:3 (1988), pp. 263-275.
Maxim Rytin, Finding the Inverse of Euler Totient Function (1999).
S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836.
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A000010, A002180, A032446, A058277.
Cf. A002110, A005277, A007614, A007617 (complement).
Cf. A083533 (first differences), A264739.
Cf. A006093 (a subsequence).

import Data.List.Ordered (insertSet)
a002202 n = a002202_list !! (n-1)
a002202_list = f [1..] (tail a002110_list) [] where
   f (x:xs) ps'@(p:ps) us
     | x < p = f xs ps' $ insertSet (a000010' x) us
     | otherwise = vs ++ f xs ps ws
     where (vs, ws) = span (<= a000010' x) us
-- Reinhard Zumkeller, Nov 22 2015

with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 1 to 300 do if t1[n] <> 0 then t2 := [op(t2), n]; fi; od: t2;
# second Maple program:
q:= n-> is(numtheory[invphi](n)<>[]):
select(q, [$1..176])[];  # Alois P. Heinz, Nov 13 2024

phiQ[m_] := Select[Range[m+1, 2m*Product[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1 ] != {}; Select[Range[176], phiQ] (* Jean-François Alcover, May 23 2011, after Maxim Rytin *)

lst(lim)=my(P=1,q,v);forprime(p=2,default(primelimit), if(eulerphi(P*=p)>=lim,q=p;break));v=vecsort(vector(P/q*lim\eulerphi(P/q),k,eulerphi(k)),,8);select(n->n<=lim,v) \\ Charles R Greathouse IV, Apr 16 2012

select(istotient,vector(100,i,i)) \\ Charles R Greathouse IV, Dec 28 2012

A039770 Numbers k such that phi(k) is a perfect square.

Original entry on oeis.org

1, 2, 5, 8, 10, 12, 17, 32, 34, 37, 40, 48, 57, 60, 63, 74, 76, 85, 101, 108, 114, 125, 126, 128, 136, 160, 170, 185, 192, 197, 202, 204, 219, 240, 250, 257, 273, 285, 292, 296, 304, 315, 364, 370, 380, 394, 401, 432, 438, 444, 451, 456, 468, 489, 504, 505

Offset: 1

Olivier Gérard

A004171 is a subsequence because phi(2^(2k+1)) = (2^k)^2. - Enrique Pérez Herrero, Aug 25 2011
Subsequence of primes is A002496 since in this case phi(k^2+1) = k^2. - Bernard Schott, Mar 06 2023
Products of distinct terms of A002496 form a subsequence. - Chai Wah Wu, Aug 22 2025

			phi(34) = 16 = 4*4.

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 141.

T. D. Noe, Table of n, a(n) for n = 1..10000
W. D. Banks, J. B. Friedlander, C. Pomerance and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
P. Pollack and C. Pomerance, Square values of Euler's function, submitted for publication.
Bernard Schott, Subfamilies and subsequences

Cf. A000010, A007614. A062732 gives the squares. A306882 (squares not totient).
Cf. A068560, A114063, A078164.
Subsequences: A054755, A002496, A004171, A262406, A324745, A324746, A247129, A324747, A306908.

with(numtheory); isA039770 := proc (n) return issqr(phi(n)) end proc; seq(`if`(isA039770(n), n, NULL), n = 1 .. 505); # Nathaniel Johnston, Oct 09 2013

Select[ Range[ 600 ], IntegerQ[ Sqrt[ EulerPhi[ # ] ] ]& ]

for(n=1, 120, if (issquare(eulerphi(n)), print1(n, ", ")))

from math import isqrt
from sympy import totient as phi
def ok(n): return isqrt(p:=phi(n))**2 == p
print([k for k in range(1, 506) if ok(k)]) # Michael S. Branicky, Aug 17 2025

a(n) seems to be asymptotic to c*n^(3/2) with 1 < c < 1.3. - Benoit Cloitre, Sep 08 2002

Banks, Friedlander, Pomerance, and Shparlinski show that a(n) = O(n^1.421). - Charles R Greathouse IV, Aug 24 2009

A032447 Inverse function of phi( ).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 10, 12, 7, 9, 14, 18, 15, 16, 20, 24, 30, 11, 22, 13, 21, 26, 28, 36, 42, 17, 32, 34, 40, 48, 60, 19, 27, 38, 54, 25, 33, 44, 50, 66, 23, 46, 35, 39, 45, 52, 56, 70, 72, 78, 84, 90, 29, 58, 31, 62, 51, 64, 68, 80, 96, 102, 120, 37, 57, 63, 74, 76, 108, 114, 126

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net)

Arrange integers in order of increasing phi value; the phi values themselves form A007614.
Inverse of sequence A064275 considered as a permutation of the positive integers. - Howard A. Landman, Sep 25 2001
In the array shown in the example section row no. n gives exactly the N values for which the cyclotomic polynomials cyclotomic(N,x) have degree A002202(n). - Wolfdieter Lang, Feb 19 2012.

			phi(1)=phi(2)=1, phi(3)=phi(4)=phi(6)=2, phi(5)=phi(8)=...=4, ...
From _Wolfdieter Lang_, Feb 19 2012: (Start)
Read as array a(n,m) with row length l(n):=A058277(v(n)) with v(n):= A002202(n), n>=1. a(n,m) = m-th element of the set {m from positive integers: phi(m)=v(n)} when read as an increasingly ordered list.
  l(n): 2, 3, 4, 4, 5, 2, 6, 6, 4, 5, ...
   n, v(n)\m 1  2  3  4  5  6  7  8  9  10 11  12  13  14
   1,  1:    1  2
   2,  2:    3  4  6
   3,  4:    5  8 10 12
   4,  6:    7  9 14 18
   5,  8:   15 16 20 24 30
   6, 10:   11 22
   7, 12:   13 21 26 28 36 42
   8, 16:   17 32 34 40 48 60
   9, 18:   19 27 38 54
  10, 20:   25 33 44 50 66
  ...
Row no. n=4: The cyclotomic polynomials cyclotomic(N,x) with values N = 7,9,14, and 18 have degree 6, and only these.
(End)

Sivaramakrishnan, The many facets of Euler's Totient, I. Nieuw Arch. Wisk. 4 (1986), 175-190.

T. D. Noe, Table of n, a(n) for n = 1..10000 (Corrected by _Dana Jacobsen_, Mar 04 2019)
D. Bressoud, CNT.m Computational Number Theory Mathematica package.
H. Gupta, Euler’s totient function and its inverse, Indian J. pure appl. Math., 12(1): 22-29(1981).
Index entries for sequences that are permutations of the natural numbers

Cf. A000010, A007614.

Cf. A002110, A064275.

import Data.List.Ordered (insertBag)
a032447 n = a032447_list !! (n-1)
a032447_list = f [1..] a002110_list [] where
   f xs'@(x:xs) ps'@(p:ps) us
     | x < p = f xs ps' $ insertBag (a000010' x, x) us
     | otherwise = map snd vs ++ f xs' ps ws
     where (vs, ws) = span ((<= a000010' x) . fst) us
-- Reinhard Zumkeller, Nov 22 2015

Needs["CNT`"]; Flatten[Table[PhiInverse[n], {n, 40}]] (* T. D. Noe, Oct 15 2012 *)
Take[Values@ PositionIndex@ Array[EulerPhi, 10^3], 15] // Flatten (* Michael De Vlieger, Dec 29 2017 *)
SortBy[Table[{n,EulerPhi[n]},{n,150}],Last][[All,1]] (* Harvey P. Dale, Oct 11 2019 *)

M = 9660; /* choose a term of A036913 */
v = vector(M, n, [eulerphi(n),n] );
v = vecsort(v, (x, y)-> if( x[1]-y[1]!=0, sign(x[1]-y[1]), sign(x[2]-y[2]) ) );
P=eulerphi(M);
v = select( x->(x[1]<=P), v );
/* A007614 = vector(#v,n, v[n][1] ) */
A032447 = vector(#v,n, v[n][2] )
/* for (n=1,#v, print(n," ", A032447[n]) ); */ /* b-file */
/* Joerg Arndt, Oct 06 2012 */

use ntheory ":all"; my($n,$k,$i,@v)=(10000,1,0); push @v,inverse_totient($k++) while @v<$n; $#v=$n-1; say ++$i," $" for @v; # _Dana Jacobsen, Mar 04 2019

Example corrected, more terms and program from Olivier Gérard, Feb 1999

A058277 Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.

Original entry on oeis.org

2, 3, 4, 4, 5, 2, 6, 6, 4, 5, 2, 10, 2, 2, 7, 8, 9, 4, 3, 2, 11, 2, 2, 3, 2, 9, 8, 2, 2, 17, 2, 10, 2, 6, 6, 3, 17, 4, 2, 3, 2, 9, 2, 6, 3, 17, 2, 9, 2, 7, 2, 2, 3, 21, 2, 2, 7, 12, 4, 3, 2, 12, 2, 8, 2, 10, 4, 2, 21, 2, 2, 8, 3, 4, 2, 3, 19, 5, 2, 8, 2, 2, 6, 2, 31, 2, 9, 10

Offset: 1

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001

Carmichael (1922) conjectured that the number 1 never appears in this sequence. Sierpiński conjectured and Ford (1998) proved that all integers greater than 1 occur in the sequence. Erdős (1958) proved that if s >= 1 appears in the sequence then it appears infinitely often. - Nick Hobson, Nov 04 2006

A002202(n) occurs a(n) times in A007614. - Reinhard Zumkeller, Nov 22 2015

Édouard Lucas, Théorie des Nombres, Blanchard 1958.

T. D. Noe, Table of n, a(n) for n=1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
R. D. Carmichael, Note on Euler's totient function, Bull. Amer. Math. Soc. 28 (1922), pp. 109-110.
Paul Erdős, Some remarks on Euler's totient function, Acta Arith. 4 (1958), pp. 10-19.
M. Farrokhi D. G., Gap function to compute the inverse of Euler's totient function.
Kevin Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Nick Hobson, Solution to puzzle 152: Totient valence.
Eric Weisstein's World of Mathematics, Totient Valence Function.

The nonzero terms of A014197.
Cf. A000010, A002202, A007614.
Cf. A006511 (largest k for which A000010(k) = A002202(n)).

import Data.List (group)
a058277 n = a058277_list !! (n-1)
a058277_list = map length $ group a007614_list
-- Reinhard Zumkeller, Nov 22 2015

max = 300; inversePhi[?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m * Times @@ (p/(p-1)); n = m; nn = Reap[While[n <= nmax, If[EulerPhi[n] == m, Sow[n]]; n++]] // Last; If[nn == {}, {}, First[nn] ] ]; Reap[For[n = 1, n <= max, n = If[n == 1, 2, n+2], nn = inversePhi[n] ; If[nn != {} , Sow[nn // Length] ] ] ] // Last // First (* Jean-François Alcover, Nov 21 2013 *)

lista(nmax) = {my(m); for(n = 1, nmax, m = invphiNum(n); if(m > 0, print1(m, ", ")));} \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

More terms from Nick Hobson, Nov 04 2006

A039771 Numbers k such that phi(k) is a perfect cube.

Original entry on oeis.org

1, 2, 15, 16, 20, 24, 30, 85, 128, 136, 160, 170, 192, 204, 240, 247, 259, 327, 333, 351, 399, 405, 436, 494, 518, 532, 648, 654, 666, 684, 702, 756, 771, 798, 810, 1024, 1028, 1088, 1111, 1255, 1280, 1360, 1375, 1536, 1542, 1632, 1843, 1853, 1875, 1920, 2008

Offset: 1

Olivier Gérard

a(n) is prime only for a(2)=2, for other cases: eulerphi(p) = p-1 = n^3, and p = 1 + n^3 = (n+1)(n^2-n+1), so p cannot be a prime. - Enrique Pérez Herrero, Aug 29 2010

A013730 is a subsequence. - Enrique Pérez Herrero, Aug 29 2010

			phi(247) = 216 = 6*6*6.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from E. Pérez Herrero)

Cf. A000010, A007614, A039770, A216412.

q:= n-> (c-> iroot(c, 3)^3=c)(numtheory[phi](n)):
select(q, [$1..2200])[];  # Alois P. Heinz, Sep 01 2025

Select[ Range[ 2000 ], IntegerQ[ Power[ EulerPhi[ # ], 1/3 ] ]& ]

for(n=1,1e4,if(ispower(eulerphi(n),3),print1(n", "))) \\ Charles R Greathouse IV, Jul 31 2011

A039698 Numbers k such that phi(k) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 46, 47, 48, 49, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 79, 82, 83, 86, 88, 89, 91, 93, 94, 95, 97, 98, 99, 100, 101, 103

Offset: 1

Olivier Gérard

Positive integers k for which values of A039649(k) are primes. - Vladimir Shevelev, May 10 2008
For every prime p, the numbers p and 2p are terms of this sequence. - Vladimir Shevelev, May 10 2008
Union of A000040 and A066071. - Ray Chandler, May 26 2008

			phi(10)+1 = 4+1 = 5, a prime number, so 10 is a term.

Antti Karttunen, Table of n, a(n) for n = 1..26197 (first 1000 terms from Vincenzo Librandi)

Cf. A000010, A000040, A006093, A039649, A066071, A007614.
Cf. A039689 (complement), A296079 (characteristic function).
Cf. also A065512, A078892, A263028, A248792.

[n: n in [1..200] | IsPrime(EulerPhi(n)+1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[300], PrimeQ[EulerPhi[#] + 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A085713 Consider numbers k such that phi(x) = k has exactly 3 solutions and they are (3p, 4p, 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

Alford Arnold, Jul 19 2003

nonn

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

			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).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from Reinhard Zumkeller)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
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.

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

Cf. A007614, A010051.

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

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]

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

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

A035113 Numbers != 2 (mod 4) listed in order of increasing totient function phi (A000010).

Original entry on oeis.org

1, 3, 4, 5, 8, 12, 7, 9, 15, 16, 20, 24, 11, 13, 21, 28, 36, 17, 32, 40, 48, 60, 19, 27, 25, 33, 44, 23, 35, 39, 45, 52, 56, 72, 84, 29, 31, 51, 64, 68, 80, 96, 120, 37, 57, 63, 76, 108, 41, 55, 75, 88, 100, 132, 43, 49, 69, 92, 47, 65, 104, 105, 112, 140, 144

Offset: 1

N. J. A. Sloane

			phi(1)=1, phi(3)=2, phi(4)=2, phi(5)=4, ...

Dumitru Damian, Table of n, a(n) for n = 1..10000
L. C. Washington, The Main Conjecture and Annihilation of Class Groups, In: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol 83 (1997) Springer, New York, NY.

Cf. A000010, A035114.

Cf. A002181, A002202, A007614, A014197, A016825, A032447, A058277.

from sympy import totient as A000010
def lov(n): return sorted([[A000010(n), n] for n in range(1,n) if n%4 != 2])
print([x[1] for x in lov(200)][:100]) # Dumitru Damian, Feb 01 2022

More terms from James Sellers

a(43) onward corrected by Sean A. Irvine, Sep 26 2020

A035114 Values of phi(n) corresponding to A035113.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 12, 12, 12, 12, 16, 16, 16, 16, 16, 18, 18, 20, 20, 20, 22, 24, 24, 24, 24, 24, 24, 24, 28, 30, 32, 32, 32, 32, 32, 32, 36, 36, 36, 36, 36, 40, 40, 40, 40, 40, 40, 42, 42, 44, 44, 46, 48, 48, 48, 48, 48, 48, 48, 48, 48

Offset: 1

N. J. A. Sloane

			phi(1)=1, phi(3)=2, phi(4)=2, phi(5)=4, ...

Dumitru Damian, Table of n, a(n) for n = 1..10000
L. C. Washington, The Main Conjecture and Annihilation of Class Groups, In: Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol 83 (1997) Springer, New York, NY.

Cf. A000010, A035113.

Cf. A002181, A002202, A007614, A014197, A016825, A032447, A058277.

from sympy import totient as A000010
def lov(n): return sorted([[A000010(n), n] for n in range(1,n) if n%4 != 2])
print([x[0] for x in lov(200)][:100]) # Dumitru Damian, Feb 03 2022

a(n) = A000010(A035113(n)). - Michel Marcus, Feb 07 2022

More terms from James Sellers

a(43) onward corrected by Sean A. Irvine, Sep 26 2020

A039689 Numbers k such that phi(k) + 1 is not a prime.

Original entry on oeis.org

15, 16, 20, 24, 25, 30, 33, 35, 39, 44, 45, 50, 51, 52, 56, 64, 65, 66, 68, 69, 70, 72, 78, 80, 81, 84, 85, 87, 90, 92, 96, 102, 104, 105, 112, 116, 120, 121, 123, 128, 129, 130, 136, 138, 140, 141, 143, 144, 147, 155, 156, 159, 160, 161, 162, 164, 165, 168, 170

Offset: 1

Olivier Gérard

			phi(20)+1 = 8+1 = 9 is not prime.

Antti Karttunen, Table of n, a(n) for n = 1..39340

Cf. A000010, A007614, A039649, A039698 (complement).
Positions of zeros in A296079.
Cf. also A263029.

Select[Range[200],!PrimeQ[EulerPhi[#]+1]&] (* Harvey P. Dale, Aug 31 2018 *)

isok(k) = !isprime(eulerphi(k)+1); \\ Michel Marcus, Jun 28 2021

Name edited by Antti Karttunen, Dec 05 2017

cp's OEIS Frontend

A002202 Values taken by totient function phi(m) (A000010).

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A039770 Numbers k such that phi(k) is a perfect square.

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A032447 Inverse function of phi( ).

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A058277 Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.

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A039771 Numbers k such that phi(k) is a perfect cube.

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A039698 Numbers k such that phi(k) + 1 is prime.

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A085713 Consider numbers k such that phi(x) = k has exactly 3 solutions and they are (3p, 4p, 6*p) where p is 1 or a prime. Sequence gives values of p.