A071386 -id:A071386 - 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)

A071387 Smallest number k for which the set of solutions to phi(x) = k has 2n-1 entries.

Original entry on oeis.org

0, 2, 8, 32, 40, 48, 396, 704, 72, 216, 144, 1056, 360, 384, 1320, 240, 9000, 7128, 480, 1296, 15936, 3072, 864, 7344, 720, 4992, 2016, 6000, 4752, 3024, 9984, 1920, 7560, 22848, 29160, 3360, 13248, 27720, 9072, 9360, 4032, 4800, 16896, 3840, 9504, 23520, 5040

Offset: 1

Labos Elemer, May 23 2002

nonn

			For n = 7: 2n-1 = 13, a(7) = Min(InvPhi(13)) = Min({396,400,420,552,560,660}) = 396.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (see invphi.gp there).

Cf. A000010 (phi), A014197, A058277, A063512, A071386, A071388, A071389.

a(n) = {if (n==1, return (0)); my(k=1); while(#invphi(k) != 2*n-1, k++); k;} \\ Michel Marcus, May 13 2020

a(n) = Min({x; Card(InvPhi(x)) = 2n-1}); a(1)=0 because of Carmichael conjecture.

a(12)-a(47) from Donovan Johnson, Jul 27 2011

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

Labos Elemer, May 23 2002

nonn

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

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

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

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

filter:= n -> isprime(nops(numtheory:-invphi(n))):
select(filter, [$1..400]); # Robert Israel, Mar 29 2020

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

A071389 Least number m such that cardinality of InvPhi(m) = prime(n).

Original entry on oeis.org

1, 2, 8, 32, 48, 396, 72, 216, 1056, 1320, 240, 480, 15936, 3072, 7344, 2016, 3024, 9984, 22848, 3360, 13248, 9360, 4800, 9504, 9216, 23328, 7680, 53280, 12480, 29376, 91200, 159744, 22464, 228960, 29952, 179200, 47040, 68544, 15840, 20736, 61440

Offset: 1

Labos Elemer, May 23 2002

nonn

			For n = 11: prime(11) = 31, Card(InvPhi(x)) = 31 for {240, 672, ...}; the smallest is 240 = a(11).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A014197, A058277, A063512, A071386, A071387, A071388.

lista(len) = {my(p = prime(len), v = vector(p, i, -!isprime(i)), c = 0, k = 1, i); while(c < len, i = invphiNum(k); if(i > 0 && i <= p && v[i] == 0, c++; v[i] = k); k++); select(x -> x > 0, v);} \\ Amiram Eldar, Nov 11 2024, using Max Alekseyev's invphi.gp

a(n) = Min{x; Card(InvPhi(x)) = prime(n), n-th prime}

4 more terms from Emeric Deutsch, Jul 25 2005

More terms from Max Alekseyev, Apr 24 2010

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

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

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A071387 Smallest number k for which the set of solutions to phi(x) = k has 2n-1 entries.

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

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A071389 Least number m such that cardinality of InvPhi(m) = prime(n).

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PARI

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

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Mathematica

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