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A014197
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Number of numbers m with Euler phi(m) = n.
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30
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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
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1,1
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COMMENTS
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Carmichael conjectured that there are no 1's in this sequence.
Number of cyclotomic polynomials of degree n. - T. D. Noe, Aug 15 2003
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, section B39.
J. Roberts, Lure of The Integers, entry 32, page 182.
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LINKS
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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
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FORMULA
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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
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MAPLE
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with(numtheory): A014197 := n-> nops(invphi(i));
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MATHEMATICA
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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}] (* From Jean-François Alcover, Dec 09 2011 *)
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PROG
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(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]
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CROSSREFS
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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
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Additional comments from Jud McCranie, Oct 10 2000
Replaced a geocities.com URL - R. J. Mathar, Oct 30 2009
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STATUS
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approved
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