

A002202


Values taken by totient function phi(m) (A000010).
(Formerly M0987 N0371)


59



1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176
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OFFSET

1,2


COMMENTS

These are the numbers n such that for some m the multiplicative group mod m has order n.
Maier & Pomerance show that there are about x * exp(c (log log log x)^2)/log x members of this sequence up to x, with c = 0.81781465... (A234614); see the paper for details on making this precise.  Charles R Greathouse IV, Dec 28 2013


REFERENCES

J. W. L. Glaisher, NumberDivisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 2734.
Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phifunction, Acta Arithmetica 49:3 (1988), pp. 263275.
Eric Weisstein's World of Mathematics, Totient Valence Function


MAPLE

with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 1 to 300 do if t1[n] <> 0 then t2 := [op(t2), n]; fi; od: t2;


MATHEMATICA

phiQ[m_] := Select[Range[m+1, 2m*Product[(11/(k*Log[k]))^(1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1 ] != {}; Select[Range[176], phiQ] (* JeanFrançois Alcover, May 23 2011, after Maxim Rytin *)


PROG

(PARI) lst(lim)=my(P=1, q, v); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); select(n>n<=lim, v) \\ Charles R Greathouse IV, Apr 16 2012
(PARI) select(istotient, vector(100, i, i)) \\ Charles R Greathouse IV, Dec 28 2012


CROSSREFS

Cf. A000010, A002180, A032446, A058277.
Sequence in context: A011860 A049445 A002174 * A049225 A076450 A097379
Adjacent sequences: A002199 A002200 A002201 * A002203 A002204 A002205


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane.


STATUS

approved



