A002202 Values taken by totient function phi(m) (A000010).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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

Crossrefs

Cf. A000010, A002180, A032446, A058277.

Cf. A002110, A005277, A007614, A007617 (complement).

Cf. A083533 (first differences), A264739.

Cf. A006093 (a subsequence).

Programs

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

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;
# second Maple program:
q:= n-> is(numtheory[invphi](n)<>[]):
select(q, [$1..176])[];  # Alois P. Heinz, Nov 13 2024

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 *)

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