A057826 -id:A057826 - cp's OEIS frontend

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

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)

A006511 Largest inverse of totient function (A000010): a(n) is the largest x such that phi(x) = m, where m = A002202(n) is the n-th number in the range of phi.

Original entry on oeis.org

2, 6, 12, 18, 30, 22, 42, 60, 54, 66, 46, 90, 58, 62, 120, 126, 150, 98, 138, 94, 210, 106, 162, 174, 118, 198, 240, 134, 142, 270, 158, 330, 166, 294, 276, 282, 420, 250, 206, 318, 214, 378, 242, 348, 354, 462, 254, 510, 262, 414, 274, 278, 426, 630, 298, 302

Offset: 1

N. J. A. Sloane

nonn

Always even, as phi(2n) = phi(n) when n is odd. - Alain Jacques (thegentleway(AT)bigpond.com), Jun 15 2006

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
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
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Richard K. Guy, Letter to N. J. A. Sloane, Jun 1991.

Cf. A000010, A002202, A002181, A057826.

For records see A036913, A132154, A036912.

phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl=={}, Return[If[n==1, {1}, {}]]]; val={}; p=Last[pl]; For[e=0; pe=1, e==0||Mod[n, (p-1)pe/p]==0, e++; pe*=p, val=Join[val, pe*phiinv[If[e==0, n, n*p/pe/(p-1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1+Divisors[n], PrimeQ]]; Last/@Select[phiinv/@Range[1, 200], #!={}&] (* phiinv[n, pl] = list of x with phi(x)=n and all prime divisors of x in list pl. phiinv[n] = list of x with phi(x)=n *)

g(n) = if(n%2, 2*(n==1), forstep(k = floor(exp(Euler)*n*log(log(n^2))+2.5*n/log(log(n^2))), n, -1, if(eulerphi(k)==n, return(k)); if(k==n, return(0)))); \\ A057635
lista(nn) = for(m = 1, nn, if(istotient(m), print1(g(m), ", "))); \\ Jinyuan Wang, Aug 29 2019

lista(nmax) = my(s); for(n = 1, nmax, s = invphiMax(n); if(s > 0, print1(s, ", "))); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

use ntheory ":all"; my $k=1; for my $i (1..100) { my @v; do{@v=inverse_totient($k++)} until @v; print "$i $v[-1]\n"; } # Dana Jacobsen, Mar 04 2019

a(n) = A057635(A002202(n)). - T. D. Noe

A066080 Largest solution x to phi(x) + 1 = prime(n).

Original entry on oeis.org

2, 6, 12, 18, 22, 42, 60, 54, 46, 58, 62, 126, 150, 98, 94, 106, 118, 198, 134, 142, 270, 158, 166, 276, 420, 250, 206, 214, 378, 348, 254, 262, 274, 278, 298, 302, 474, 486, 334, 346, 358, 594, 382, 840, 394, 398, 422, 446, 454, 458, 708, 478, 1050, 502, 1020

Offset: 1

Labos Elemer, Dec 03 2001

nonn

			For p = 97: x = {97, 119, 153, 194, 195, 208, 224, 238, 260, 280, 288, 306, 312, 336, 360, 390, 420} is the set of 17 solutions such that phi(x) + 1 = 97.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
David M. Bressoud, A Course in Computational Number Theory (webpage), CNT.m, Computational Number Theory Mathematica package.

Cf. A000010, A057826 (greatest number with totient 2n).

Table[Do[s=1+EulerPhi[n]; If[Equal[s, Prime[k]], Print[{n, s}]], {n, 1, 4*Prime[k]^2}], {k, 1, 100}]
Needs["CNT`"]; Table[Solve[EulerPhi[x] == Prime[n] - 1, x][[-1, -1, -1]], {n, 100}] (* T. D. Noe, Nov 07 2011 *)

a(n) = invphiMax(prime(n)-1); \\ Amiram Eldar, Dec 16 2024, using Max Alekseyev's invphi.gp

A066412 Number of elements in the set phi_inverse(phi(n)).

Original entry on oeis.org

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

Offset: 1

Vladeta Jovovic, Dec 25 2001

nonn

			invphi(6) = [7, 9, 14, 18], thus a(7) = a(9) = a(14) = a(18) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Euler's totient function (see the last paragraph in section "Some values of the function")

Cf. A000010, A001055, A014197, A032447, A036913, A057826, A071181.

Cf. A070305 (positions where coincides with A000005).

Maple
```
nops(invphi(phi(n)));
```

With[{nn = 120}, Function[s, Take[#, nn] &@ Values@ KeySort@ Flatten@ Map[Function[{k, m}, Map[# -> m &, k]] @@ {#, Length@ #} &@ Lookup[s, #] &, Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, nn^2 + 10]] (* Michael De Vlieger, Jul 18 2017 *)

for(n=1,150,print1(sum(i=1,10*n,if(n-eulerphi(n)-i+eulerphi(i),0,1)),",")) \\ By the original author(s). Note: the upper limit 10*n for the search range is quite ad hoc, and is guaranteed to miss some cases when n is large enough. Cf. Wikipedia-article. - Antti Karttunen, Jul 19 2017

\\ Here is an implementation not using arbitrary limits:
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
A066412(n) = A014197(eulerphi(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = invphiNum(eulerphi(n)); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

;; A naive implementation requiring precomputed A057826:
(define (A066412 n) (if (<= n 2) 2 (let ((ph (A000010 n))) (let loop ((k (A057826 (/ ph 2))) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (= ph (A000010 k)) 1 0)))))))) ;; Antti Karttunen, Jul 18 2017

a(n) = Card( k>0 : cototient(k)=cototient(n) ) where cototient(x) = x - phi(x). - Benoit Cloitre, May 09 2002
From Antti Karttunen, Jul 18 2017: (Start)
a(n) = A014197(A000010(n)).
For all n, a(n) <= A071181(n).
(End)

A028476 Greatest k such that phi(k) = phi(n), where phi is Euler's totient function.

Original entry on oeis.org

2, 2, 6, 6, 12, 6, 18, 12, 18, 12, 22, 12, 42, 18, 30, 30, 60, 18, 54, 30, 42, 22, 46, 30, 66, 42, 54, 42, 58, 30, 62, 60, 66, 60, 90, 42, 126, 54, 90, 60, 150, 42, 98, 66, 90, 46, 94, 60, 98, 66, 120, 90, 106, 54, 150, 90, 126, 58, 118, 60, 198, 62, 126, 120, 210, 66, 134

Offset: 1

Vladeta Jovovic, Jan 12 2002

nonn

Every number in this sequence occurs at least twice. For all n > 6, a(n) > phi(n)^2 is impossible. - Alonso del Arte, Dec 31 2016

			phi(1) = 1 and phi(2) = 1 also. There is no greater k such that phi(k) = 1, so therefore a(1) = a(2) = 2.
phi(3) = phi(4) = phi(6) = 2, and there is no greater k such that phi(k) = 6, hence a(3) = a(4) = a(6) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000 (computed from the b-file of A057826 provided by T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A015126, A057826, A066412, A066659, A071181.

Table[Module[{k = (2 Boole[n <= 6]) + #^2}, While[EulerPhi@ k != #, k--]; k] &@ EulerPhi@ n, {n, 120}] (* Michael De Vlieger, Dec 31 2016 *)

a(n) = invphiMax(eulerphi(n)); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

a(1) = a(2) = 2, for n > 2, a(n) = A057826(A000010(n)/2). - Antti Karttunen, Aug 07 2017

A071181 Number of numbers k such that phi(k) divides phi(n).

Original entry on oeis.org

2, 2, 5, 5, 9, 5, 9, 9, 9, 9, 7, 9, 19, 9, 14, 14, 20, 9, 13, 14, 19, 7, 7, 14, 16, 19, 13, 19, 11, 14, 13, 20, 16, 20, 34, 19, 31, 13, 34, 20, 30, 19, 13, 16, 34, 7, 7, 20, 13, 16, 27, 34, 11, 13, 30, 34, 31, 11, 7, 20, 37, 13, 31, 27, 51, 16, 13, 27, 14, 34, 9, 34, 63, 31, 30, 31

Offset: 1

Benoit Cloitre, Jun 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2048 from Antti Karttunen computed with the help of b-file of A057826 provided by T. D. Noe)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A057826, A066412.

for(n=1,100,print1(sum(i=1,1000,if(eulerphi(n)%eulerphi(i),0,1)),","))

a(n) = {my(v = 0); fordiv(eulerphi(n), d, v += invphiNum(d)); v;}  \\ Amiram Eldar, Nov 12 2024, using Max Alekseyev's invphi.gp; edited by Max Alekseyev, Nov 16 2024

;; A naive implementation requiring precomputed A057826:
(define (A071181 n) (if (<= n 2) 2 (let ((ph (A000010 n))) (let loop ((k (A057826 (/ ph 2))) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (zero? (modulo ph (A000010 k))) 1 0)))))))) ;; Antti Karttunen, Jul 18 2017

For all n, a(n) >= A066412(n). - Antti Karttunen, Jul 17 2017

A319048 a(n) is the greatest k such that A000010(k) divides n where A000010 is the Euler totient function.

Original entry on oeis.org

2, 6, 2, 12, 2, 18, 2, 30, 2, 22, 2, 42, 2, 6, 2, 60, 2, 54, 2, 66, 2, 46, 2, 90, 2, 6, 2, 58, 2, 62, 2, 120, 2, 6, 2, 126, 2, 6, 2, 150, 2, 98, 2, 138, 2, 94, 2, 210, 2, 22, 2, 106, 2, 162, 2, 174, 2, 118, 2, 198, 2, 6, 2, 240, 2, 134, 2, 12, 2, 142, 2, 270, 2, 6, 2, 12, 2, 158, 2, 330

Offset: 1

Michel Marcus, Sep 09 2018

nonn

Sándor calls this function the totient maximum function and remarks that this function is well-defined, since a(n) can be at least 2, and cannot be greater than n^2 (when n > 6).

Antti Karttunen, Table of n, a(n) for n = 1..16384
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
József Sándor, On the Euler minimum and maximum functions, Notes on Number Theory and Discrete Mathematics, Volume 15, 2009, Number 3, Pages 1—8.

Cf. A000010 (Euler totient), A061026 (the totient minimum function).
Cf. A319068 (the analog for the sum of divisors).
Cf. A070633, also A057635, A057826, A069932, A071181.
Right border of A378638.

a[n_] := Module[{kmax = If[n <= 6, 10 n, n^2]}, For[k = kmax, True, k--, If[Divisible[n, EulerPhi[k]], Return[k]]]];
Array[a, 80] (* Jean-François Alcover, Sep 17 2018, from PARI *)

a(n) = {my(kmax = if (n<=6, 10*n, n^2)); forstep (k=kmax, 1, -1, if ((n % eulerphi(k)) == 0, return (k)););}

\\ (The first two functions could probably be combined in a smarter way):
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))))} \\ From A014197 by M. F. Hasler
A057635(n) = if(1==n,2,if((n%2),0,my(k=A014197(n),i=n); if(!k, 0, while(k, i++; if(eulerphi(i)==n, k--)); (i))));
A319048(n) = { my(m=0); fordiv(n,d, m = max(m,A057635(d))); (m); }; \\ Antti Karttunen, Sep 09 2018

a(n) = {my(d = divisors(n)); vecmax(vector(#d, i, invphiMax(d[i])));} \\ Amiram Eldar, Nov 29 2024, using Max Alekseyev's invphi.gp

a(n) = Max_{d|n} A057635(d). - Antti Karttunen, Sep 09 2018

A070305 Numbers m such that Card(k>0 : phi(k)=phi(m)) = tau(m).

Original entry on oeis.org

2, 4, 8, 10, 11, 14, 16, 23, 27, 28, 29, 31, 32, 38, 47, 53, 59, 64, 67, 71, 79, 83, 86, 100, 103, 107, 114, 125, 127, 128, 131, 136, 137, 139, 147, 149, 151, 167, 170, 172, 173, 176, 179, 191, 197, 199, 202, 211, 223, 227, 229, 235, 239, 251, 256, 263, 265, 269

Offset: 1

Benoit Cloitre, May 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Antti Karttunen)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Euler's totient function (see the last paragraph in section "Some values of the function").

Cf. A000005, A000010, A057826, A066412.

With[{nn = 300}, Function[s, DeleteCases[MapIndexed[If[DivisorSigma[0, First@ #2] == #1, First@ #2, 0] &, Take[#, nn]], 0] &@ Values@ KeySort@ Flatten@ Map[Function[{k, m}, Map[# -> m &, k]] @@ {#, Length@ #} &@ Lookup[s, #] &, Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(4/3)] + 10]] (* Michael De Vlieger, Jul 18 2017 *)

for(n=1,350,if(sum(i=1,10*n,if(eulerphi(n)-eulerphi(i),0,1))==numdiv(n),print1(n,","))) \\ By the original author. Note: the upper limit 10*n for the search range is quite ad hoc, and is guaranteed to miss some cases when n is large enough. Cf. Wikipedia-article. - Antti Karttunen, Jul 19 2017

\\ Here is an implementation not using arbitrary limits:
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

A066412(n) = A014197(eulerphi(n));
isA070305(n) = (A066412(n) == numdiv(n));
n=0; k=1; while(k <= 1000, n=n+1; if(isA070305(n),write("b070305.txt", k, " ", n);k=k+1)); \\ Antti Karttunen, Jul 19 2017

is(m) = {my(f = factor(m)); invphiNum(eulerphi(f)) == numdiv(f);} \\ Amiram Eldar, Nov 19 2024, using Max Alekseyev's invphi.gp

;; With my IntSeq-library.
(define A070305 (MATCHING-POS 1 1 (lambda (n) (= (A066412 n) (A000005 n))))) ;; Antti Karttunen, Jul 18 2017

Numbers k such that A066412(k) = A000005(k).

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

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A006511 Largest inverse of totient function (A000010): a(n) is the largest x such that phi(x) = m, where m = A002202(n) is the n-th number in the range of phi.

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A066080 Largest solution x to phi(x) + 1 = prime(n).

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A066412 Number of elements in the set phi_inverse(phi(n)).

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A028476 Greatest k such that phi(k) = phi(n), where phi is Euler's totient function.

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A071181 Number of numbers k such that phi(k) divides phi(n).

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