

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
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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/(p1)^s1/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}] (* JeanFrançois 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



