A032446 -id:A032446 - 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

A014197 Number of numbers m with Euler phi(m) = n.

Original entry on oeis.org

2, 3, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 0, 0, 6, 0, 4, 0, 5, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 7, 0, 0, 0, 8, 0, 0, 0, 9, 0, 4, 0, 3, 0, 2, 0, 11, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 9, 0, 0, 0, 8, 0, 2, 0, 0, 0, 2, 0, 17, 0, 0, 0, 0, 0, 2, 0, 10, 0, 2, 0, 6, 0, 0, 0, 6, 0, 0, 0, 3

Offset: 1

N. J. A. Sloane

Carmichael conjectured that there are no 1's in this sequence. - Jud McCranie, Oct 10 2000
Number of cyclotomic polynomials of degree n. - T. D. Noe, Aug 15 2003
Let v == 0 (mod 24), w = v + 24, and v < k < q < w, where k and q are integer. It seems that, for most values of v, there is no b such that b = a(k) + a(q) and b > a(v) + a(w). The first case where b > a(v) + a(w) occurs at v = 888: b = a(896) + a(900) = 15 + 4, b > a(888) + a(912), or 19 > 8 + 7. The first case where v < n < w and a(n) > a(v) + a(w) occurs at v = 2232: a(2240) > a(2232) + a(2256), or 27 > 7 + 8. - Sergey Pavlov, Feb 05 2017
One elementary result relating to phi(m) is that if m is odd, then phi(m)=phi(2m) because 1 and 2 both have phi value 1 and phi is multiplicative. - Roderick MacPhee, Jun 03 2017

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B39, pp. 144-146.
Joe Roberts, Lure of The Integers, The Mathematical Association of America, 1992, entry 32, page 182.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
R. D. Carmichael, On Euler’s phi-function, Bull. Amer. Math. Soc. 13 (1907), 241-243.
R. D. Carmichael, Note on Euler’s phi-function, Bull. Amer. Math. Soc. 28 (1922), 109-110.
Kevin Ford, The number of solutions of phi(x)=m, arXiv:math/9907204 [math.NT], 1999.
S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836.
Primefan, Totient Answers For The First 1000 Integers.
Eric Weisstein's World of Mathematics, Totient Function.
Eric Weisstein's World of Mathematics, Totient Valence Function.

Cf. A000010, A002202, A032446 (bisection), A049283, A051894, A055506, A057635, A057826, A058277 (nonzero terms), A058341, A063439, A066412, A070243 (partial sums), A070633, A071386 (positions of odd terms), A071387, A071388 (positions of primes), A071389 (where prime(n) occurs for the first time), A082695, A097942 (positions of records), A097946, A120963, A134269, A219930, A280611, A280709, A280712, A296655 (positions of positive even terms), A305353, A305656, A319048, A322019.
For records see A131934.
Column 1 of array A320000.

a := function(n)
local S, T, R, max, i, k, r;
S:=[];
for i in DivisorsInt(n)+1 do
    if IsPrime(i)=true then
        S:=Concatenation(S,[i]);
    fi;
od;
T:=[];
for k in [1..Size(S)] do
    T:=Concatenation(T,[S[k]/(S[k]-1)]);
od;
max := n*Product(T);
R:=[];
for r in [1..Int(max)] do
    if Phi(r)=n then
        R:=Concatenation(R,[r]);
    fi;
od;
return Size(R);
end; # Miles Englezou, Oct 22 2024

[#EulerPhiInverse(n): n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory): A014197:=n-> nops(invphi(n)): seq(A014197(n), n=1..200);

a[1] = 2; a[m_?OddQ] = 0; a[m_] := Module[{p, nmax, n, k}, p = Select[ Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; k = 0; While[n <= nmax, If[EulerPhi[n] == m, k++]; n++]; k]; Array[a, 92] (* Jean-François Alcover, Dec 09 2011, updated Apr 25 2016 *)
With[{nn = 116}, Function[s, Function[t, Take[#, nn] &@ ReplacePart[t, Map[# -> Length@ Lookup[s, #] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(3/2)] + 10]] (* Michael De Vlieger, Jul 19 2017 *)

A014197(n,m=1) = { n==1 && return(1+(m<2)); my(p,q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0,valuation(q=n\d,p=d+1), A014197(q\p^i,p))))} \\ M. F. Hasler, Oct 05 2009

a(n) = invphiNum(n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

from sympy import totient, divisors, isprime, prod
def a(m):
    if m == 1: return 2
    if m % 2: return 0
    X = (x + 1 for x in divisors(m))
    nmax=m*prod(i/(i - 1) for i in X if isprime(i))
    n=m
    k=0
    while n<=nmax:
        if totient(n)==m:k+=1
        n+=1
    return k
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 18 2017, after Mathematica code

Dirichlet g.f.: Sum_{n>=1} a(n)*n^-s = zeta(s)*Product_(1+1/(p-1)^s-1/p^s). - Benoit Cloitre, Apr 12 2003
Limit_{n->infinity} (1/n) * Sum_{k=1..n} a(k) = zeta(2)*zeta(3)/zeta(6) = 1.94359643682075920505707036... (see A082695). - Benoit Cloitre, Apr 12 2003
From Christopher J. Smyth, Jan 08 2017: (Start)
Euler transform = Product_{n>=1} (1-x^n)^(-a(n)) = g.f. of A120963.
Product_{n>=1} (1+x^n)^a(n)
= Product_{n>=1} ((1-x^(2n))/(1-x^n))^a(n)
= Product_{n>=1} (1-x^n)^(-A280712(n))
= Euler transform of A280712 = g.f. of A280611.
(End)
a(A000010(n)) = A066412(n). - Antti Karttunen, Jul 18 2017
From Antti Karttunen, Dec 04 2018: (Start)
a(A000079(n)) = A058321(n).
a(A000142(n)) = A055506(n).
a(A017545(n)) = A063667(n).
a(n) = Sum_{d|n} A008683(n/d)*A070633(d).
a(n) = A056239(A322310(n)).
(End)

A210500 Number of even solutions to phi(k) = prime(n) - 1.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 5, 2, 1, 1, 1, 5, 6, 2, 1, 1, 1, 5, 1, 1, 11, 1, 1, 4, 13, 2, 1, 1, 5, 4, 1, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 5, 1, 17, 1, 1, 1, 1, 1, 1, 4, 1, 21, 1, 9, 1, 1, 1, 5, 5, 1, 1, 1, 1, 10, 1, 1, 13, 1, 3, 9, 1, 1, 1, 1, 1, 1, 7, 9, 4, 1, 7, 1, 23, 1, 1, 9

Offset: 1

Arkadiusz Wesolowski, Jan 19 2013

nonn

a(n) >= A210501(n).

			The set {k: phi(k) = 12} is {13, 21, 26, 28, 36, 42}. Thus, if phi(k) = prime(6) - 1, the equation has exactly four even solutions. Hence, a(6) = 4.

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006, pp. 52-56.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A000010, A032446, A058339, A066080, A210501, A210502.

r = 87; lst1 = Table[EulerPhi[n], {n, (Prime[r] - 1)^2 + 2}]; lst2 = {}; Do[p = Prime[n]; AppendTo[lst2, Length@Select[Flatten@Position[Take[lst1, {p - 1, (p - 1)^2 + 2}], Prime[n] - 1], OddQ]], {n, r}]; lst2

a(n) = A058339(n) - A210501(n).

A210501 Number of odd solutions to phi(k) = prime(n) - 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 4, 1, 1, 6, 1, 1, 2, 4, 2, 1, 1, 4, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 5, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 10, 1, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 1, 1, 5, 1, 3, 3, 1, 1, 1, 1, 1, 1, 6, 4, 2, 1, 6, 1, 11, 1, 1, 3

Offset: 1

Arkadiusz Wesolowski, Jan 19 2013

nonn

a(n) <= A210500(n).

			The set {k: phi(k) = 12} is {13, 21, 26, 28, 36, 42}. Thus, if phi(k) = prime(6) - 1, the equation has exactly two odd solutions. Hence, a(6) = 2.

Alexander S. Karpenko, Lukasiewicz's Logics and Prime Numbers, Luniver Press, Beckington, 2006, pp. 52-56.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A000010, A032446, A058339, A210500, A210502.

r = 87; lst1 = Table[EulerPhi[n], {n, (Prime[r] - 1)^2 + 1}]; lst2 = {}; Do[p = Prime[n]; AppendTo[lst2, Length@Select[Flatten@Position[Take[lst1, {p - 1, (p - 1)^2 + 1}], Prime[n] - 1], EvenQ]], {n, r}]; lst2

a(n) = A058339(n) - A210500(n).

A051478 a(n) is the number of values k satisfying phi(k) = 4*n+2, n>0.

Original entry on oeis.org

4, 2, 0, 4, 2, 0, 2, 0, 0, 4, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 4, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jud McCranie

nonn

Sequence consists of only 0's, 2's and 4's.

			phi(k) = 4*1+2 has 4 solutions (k = 7, 9, 14, 18), so a(1) = 4.

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

Cf. A000010, A032446, A051479.

a(n) = invphiNum(4*n+2); \\ Amiram Eldar, Nov 05 2024, using Max Alekseyev's invphi.gp

A051479 Values of i such that phi(x) = 4i+2 has 4 solutions.

Original entry on oeis.org

1, 4, 10, 40, 121, 364, 1105, 1540, 3601, 4795, 5662, 6601, 9841, 19414, 35815, 103201, 131950, 141224, 210910, 245272, 378532, 505876, 613480, 762565, 986545, 1010527, 1147576, 1252720, 1732847, 1750990, 1766905, 1798951, 1863907, 2337352, 2674042, 2773057, 3080902

Offset: 1

Jud McCranie

nonn

			phi(x) = 4*1+2 has 4 solutions (x = 7,9,14,18), so 1 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..210
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A032446, A051478.

is(k) = invphiNum(4*k+2) == 4; \\ Amiram Eldar, Nov 05 2024, using Max Alekseyev's invphi.gp

a(33)-a(37) from Amiram Eldar, Nov 05 2024

A085758 Least m such that phi(x)=2m has exactly n solutions.

Original entry on oeis.org

5, 1, 2, 4, 6, 16, 18, 20, 12, 24, 80, 198, 1134, 352, 156, 36, 168, 108, 468, 72, 312, 528, 880, 180, 1280, 192, 144, 660, 1848, 120, 384, 4500, 216, 3564, 2100, 240, 288, 648, 600, 7968, 1656, 1536, 1620, 432, 1560, 3672, 1944, 360, 840, 2496, 8820, 1008, 576

Offset: 2

Lekraj Beedassy, Jul 22 2003

nonn

Sequence refers to the rank of the first occurrence of n in A032446.

			a(0) = 7. Carmichael conjectured that a(1) doesn't exist.

Cf. A032446.

a(n) = A007374(n)/2 for n > 2. - David Wasserman, Feb 09 2005

More terms from David Wasserman, Feb 09 2005

A172464 Numbers n such that phi(phi(n)) + sigma(sigma(n)) is a 4th power.

Original entry on oeis.org

9, 42, 101, 339, 407, 420, 471, 915, 1409, 2572, 2847, 3706, 4069, 6631, 6720, 7229, 9212, 14051, 16641, 31453, 33067, 33146, 35701, 37425, 37675, 37911, 48016, 48272, 53101, 55956, 56906, 68895, 73474, 75023, 83525, 84676, 86928, 94525, 101428, 101743, 115925

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

			phi(phi(9)) + sigma(sigma(9))= 1;
phi(phi(42)) + sigma(sigma(42))= 4^4 = 256;
phi(phi(101)) + sigma(sigma(101))= 4^4 = 256;
phi(phi(339)) + sigma(sigma(339))= 6^4 = 1296.

W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
R. K. Guy, Unsolved Problems in Number Theory, B42.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..5749
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Eric Weisstein's World of Mathematics, Totient Valence Function
Eric Weisstein's World of Mathematics, Carmichael's Totient Function conjecture

Cf. A000010, A002180, A032446, A058277.

with(numtheory): for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.25) =( phi(phi(n)) + sigma(sigma(n)))^.25 then print (n);fi ; od;

Select[Range[116000],IntegerQ[Surd[DivisorSigma[1,DivisorSigma[1,#]]+ EulerPhi[ EulerPhi[ #]],4]]&] (* Harvey P. Dale, Aug 16 2021 *)

a(40)-a(41) from Hiroaki Yamanouchi, Sep 19 2014

A172465 Numbers n such that phi(phi(n)) + sigma(sigma(n)) is an 8th power.

Original entry on oeis.org

42, 101, 6720, 9212, 226570, 276404, 288086, 299668, 339098, 392228, 412276, 423395, 530917, 535759, 559427, 564209, 666181, 2835284, 3592300, 3911744, 4080100, 5980673, 7230960, 8787900, 14960924, 17130550, 23324242, 27449729, 30437729, 33869141, 42073800

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

			phi(phi(9)) + sigma(sigma(9))= 1;
phi(phi(42)) + sigma(sigma(42))= 2^8 = 256;
phi(phi(101)) + sigma(sigma(101))= 2^8 = 256;
phi(phi(6720)) + sigma(sigma(6720))= 4^8 = 65536.

W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
S. W. Golomb, Equality among number-theoretic functions, Abstract 882-11-16, Abstracts Amer. Math. Soc., 14 (1993), 415-416.
R. K. Guy, Unsolved Problems in Number Theory, B42.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..267
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function conjecture

Cf. A000010, A002180, A032446, A058277.

with(numtheory):for n from 1 to 2000000 do;if floor(( phi(phi(n)) + sigma(sigma(n)))^.125) = (phi(phi(n)) + sigma(sigma(n)))^.125 then print (n);fi ; od;

isok(n) = ispower(eulerphi(eulerphi(n)) + sigma(sigma(n)), 8); \\ Michel Marcus, Sep 20 2014

a(10) corrected and a(18)-a(31) added by Hiroaki Yamanouchi, Sep 19 2014

A296655 Numbers k such that phi(x) = k has a positive even number of solutions.

Original entry on oeis.org

1, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, 46, 52, 54, 58, 64, 66, 70, 78, 80, 82, 84, 88, 100, 102, 106, 110, 112, 126, 130, 136, 138, 148, 150, 160, 162, 166, 168, 172, 176, 178, 180, 184, 190, 196, 198, 200, 208, 210, 222, 224, 226, 228, 232, 238, 250

Offset: 1

Torlach Rush, Dec 17 2017

nonn

When the number of solutions is 2, the sum of Sum_{d|x} d*mu(d) is always 0.

A007366 is contained in this sequence because it selects terms with the smallest even number of inverses.

			1 is a term because phi(1) has two inverses, 1, and 2.
6 is a term because phi(6) has four inverses, 7, 9, 14, and 18.
10 is a term because phi(10) has two inverses, 11, and 22.
18 is a term because phi(18) has four inverses, 19, 27, 38, 54.
348 is a term because phi(348) has six inverses, 349, 413, 531, 698, 826, and 1062.

Cf. A007366, A014197, A032446, A071386.

With[{nn = 1500}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], EvenQ@ Last@ # &][[All, 1]], # <= nn/6 &] ] (* Michael De Vlieger, Dec 20 2017 *)

0 = card({phi^-1(a(n))}) mod 2.

Corrected and extended by Rémy Sigrist, Dec 19 2017

cp's OEIS Frontend

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

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A014197 Number of numbers m with Euler phi(m) = n.

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A210500 Number of even solutions to phi(k) = prime(n) - 1.

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A210501 Number of odd solutions to phi(k) = prime(n) - 1.

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A051478 a(n) is the number of values k satisfying phi(k) = 4*n+2, n>0.

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A051479 Values of i such that phi(x) = 4i+2 has 4 solutions.

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A085758 Least m such that phi(x)=2m has exactly n solutions.

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