login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A014197 Number of numbers m with Euler phi(m) = n. 30
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

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

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

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 6 06:58 EST 2016. Contains 278775 sequences.