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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.

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, [math/9907204] The number of solutions of phi(x)=m

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)*prod(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... - Benoit Cloitre, Apr 12 2003

MAPLE

with(numtheory): A014197 := n-> nops(invphi(i));

MATHEMATICA

inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[ {p, nmax, n, nn}, p = Select[ Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; nn = {}; While[n <= nmax, If[ EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; a[n_] := Length[ inversePhi[n] ]; Table[ a[n], {n, 1, 92}] (* Jean-Fran├žois Alcover, Dec 09 2011 *)

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))))} [From M. F. Hasler, Oct 05 2009]

CROSSREFS

Cf. A058277, A002202, A032446.

Cf. A070243 (partial sums).

For records see A131934, A097942.

Sequence in context: A122059 A164917 A166238 * A181308 A021438 A195822

Adjacent sequences:  A014194 A014195 A014196 * A014198 A014199 A014200

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Additional comments from Jud McCranie, Oct 10 2000

STATUS

approved

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Last modified April 16 00:30 EDT 2014. Contains 240534 sequences.