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A014197 Number of numbers m with Euler phi(m) = n. 33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, section B39.

J. Roberts, Lure of The Integers, entry 32, page 182.

LINKS

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

K. 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

FORMULA

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

Lim_{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)

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Dec 09 2011, updated Apr 25 2016 *)

PROG

(PARI) 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

CROSSREFS

Cf. A002202, A032446, A058277, A070243 (partial sums), A082695, A097942.

For records see A131934.

Sequence in context: A122059 A164917 A166238 * A181308 A277141 A021438

Adjacent sequences:  A014194 A014195 A014196 * A014198 A014199 A014200

KEYWORD

nonn,nice,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified January 24 08:32 EST 2017. Contains 281227 sequences.