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 A002202 Values taken by totient function phi(m) (A000010). (Formerly M0987 N0371) 109
 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 (list; graph; refs; listen; history; text; internal format)
 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 A264739(a(n)) = 1; a(n) occurs A058277(n) times in A007614. - Reinhard Zumkeller, Nov 26 2015 There are no odd numbers > 2 in the sequence and the even numbers that are not in the sequence are in A005277. - Bernard Schott, May 13 2020 REFERENCES J. W. L. Glaisher, Number-Divisor 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 André Contiero, and Davi Lima, 2-Adic Stratification of Totients, arXiv:2005.05475 [math.NT], 2020. K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34. Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phi-function, Acta Arithmetica 49:3 (1988), pp. 263-275. Maxim Rytin, Finding the Inverse of Euler Totient Function (1999). S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836. 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[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1 ] != {}; Select[Range[176], phiQ] (* Jean-François 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 (Haskell) import Data.List.Ordered (insertSet) a002202 n = a002202_list !! (n-1) a002202_list = f [1..] (tail a002110_list) [] where    f (x:xs) ps'@(p:ps) us      | x < p = f xs ps' \$ insertSet (a000010' x) us      | otherwise = vs ++ f xs ps ws      where (vs, ws) = span (<= a000010' x) us -- Reinhard Zumkeller, Nov 22 2015 CROSSREFS Cf. A000010, A002180, A032446, A058277. Cf. A002110, A005277, A007614, A007617 (complement). Cf. A083533 (first differences), A264739. Sequence in context: A259278 A049445 A002174 * A049225 A076450 A097379 Adjacent sequences:  A002199 A002200 A002201 * A002203 A002204 A002205 KEYWORD nonn,nice,changed AUTHOR STATUS approved

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Last modified May 24 19:05 EDT 2020. Contains 334580 sequences. (Running on oeis4.)