A320000 -id:A320000 - cp's OEIS frontend

A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

3, 6, 1, 10, 1, 12, 1, 14, 1, 4, 1, 28, 1, 1, 1, 22, 1, 12, 1, 20, 1, 4, 1, 102, 1, 1, 1, 4, 1, 4, 1, 26, 1, 1, 1, 66, 1, 1, 1, 104, 1, 12, 1, 6, 1, 4, 1, 92, 1, 1, 1, 4, 1, 4, 1, 6, 1, 4, 1, 132, 1, 1, 1, 34, 1, 4, 1, 1, 1, 4, 1, 1240, 1, 1, 1, 1, 1, 4, 1, 57, 1, 4, 1, 21, 1, 1, 1, 28, 1, 1, 1, 6, 1, 1, 1, 492, 1, 1, 1, 12, 1, 4, 1, 6, 1

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A014197, A320000, A322311 (rgs-transform).

Cf. also A322312.

A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ From A320000
A322310(n) = if(1==n,3,my(m=1); fordiv(n,d, my(s, p=d+1, q=n/d); if(isprime(p) && (s = sum(i=0,valuation(n, p), A320000sq(q/(p^i),p))), m *= prime(s))); (m));

a(n) = Product_{d|n} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)])^A010051(1+d).

For all n, A056239(a(n)) = A014197(n).

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)

A305058 Totients t such that the number of divisors of t equals the number of solutions of phi(x) = t.

Original entry on oeis.org

6, 12, 80, 160, 312, 352, 928, 1760, 1792, 3264, 3960, 7104, 7648, 13680, 15984, 16224, 17760, 19712, 20352, 20800, 21088, 22368, 23184, 25728, 25888, 26240, 27072, 29664, 47952, 57312, 60048, 62976, 67072, 73152, 74368, 77664, 78144, 81568, 85056, 85392, 86688

Offset: 1

Torlach Rush, May 24 2018

nonn

For known terms gcd({phi(x) = t}) = 1.
This is not always the case, the smallest counterexample being t=4598784, a term of A303745, which has gcd({phi(x) = t}) = 1997. - Daniel Suteu, Dec 01 2018
Conjecture: Every term divides one or more subsequent terms.
Numbers n for which A000005(n) = A014197(n), positions of zeros in A322019. - Antti Karttunen, Dec 01 2018

			6 is a term because the divisors of 6 are {1,2,3,6} and the solutions of phi(x) = 6 are {7,9,14,18}.
12 is a term because the divisors of 12 are {1,2,3,4,6,12} and the solutions of phi(x) = 12 are {13,21,26,28,36,42}.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A014197, A320000, A322019, A303745.

A014197[1] = 2; A014197[m_?OddQ] = 0; A014197[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]; aQ[n_] := (DivisorSigma[0 ,n] == A014197[n]); Select[Range[1000], aQ] (* Amiram Eldar, Dec 02 2018 after Jean-François Alcover at A014197 *)

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
isA305058(n) = (numdiv(n) == A014197(n)); \\ Antti Karttunen, Dec 01 2018

use ntheory ':all'; for (1..10**5) { print "$\n" if inverse_totient($) == divisor_sum($, 0) } # _Daniel Suteu, Dec 01 2018

tau(a(n)) = #{phi(x) = a(n)}.

More terms from Daniel Suteu, Dec 01 2018

A322311 Lexicographically earliest such sequence a that a(i) = a(j) => A322310(i) = A322310(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 3, 3, 9, 3, 5, 3, 10, 3, 7, 3, 11, 3, 3, 3, 7, 3, 7, 3, 12, 3, 3, 3, 13, 3, 3, 3, 14, 3, 5, 3, 2, 3, 7, 3, 15, 3, 3, 3, 7, 3, 7, 3, 2, 3, 7, 3, 16, 3, 3, 3, 17, 3, 7, 3, 3, 3, 7, 3, 18, 3, 3, 3, 3, 3, 7, 3, 19, 3, 7, 3, 20, 3, 3, 3, 8, 3, 3, 3, 2, 3, 3, 3, 21, 3, 3, 3, 5, 3, 7, 3, 2, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of A322310.

For all i, j: a(i) = a(j) => A014197(i) = A014197(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A014197, A320000, A322310, A322313.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A320000sq(n, k) = if(1==n, if(1==k,2,1), sumdiv(n, d, if(d>=k && isprime(d+1), my(p=d+1, q=n/d); sum(i=0, valuation(n, p), A320000sq(q/(p^i), p))))); \\ Cf. A320000
A322310(n) = if(1==n,3,my(m=1); fordiv(n,d, my(s, p=d+1, q=n/d); if(isprime(p) && (s = sum(i=0,valuation(n, p), A320000sq(q/(p^i),p))), m *= prime(s))); (m));
v322311 = rgs_transform(vector(up_to, n, A322310(n)));
A322311(n) = v322311[n];

cp's OEIS Frontend

A322310 a(n) = Product_{d|n, d+1 is prime} A008578(1+[Sum_{i=0..A286561(n,1+d)} A320000((n/d)/((1+d)^i), 1+d)]). Here A286561(n,k) gives the k-valuation of n (for k > 1).

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

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A305058 Totients t such that the number of divisors of t equals the number of solutions of phi(x) = t.

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A322311 Lexicographically earliest such sequence a that a(i) = a(j) => A322310(i) = A322310(j) for all i, j.

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