

A058277


Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.


20



2, 3, 4, 4, 5, 2, 6, 6, 4, 5, 2, 10, 2, 2, 7, 8, 9, 4, 3, 2, 11, 2, 2, 3, 2, 9, 8, 2, 2, 17, 2, 10, 2, 6, 6, 3, 17, 4, 2, 3, 2, 9, 2, 6, 3, 17, 2, 9, 2, 7, 2, 2, 3, 21, 2, 2, 7, 12, 4, 3, 2, 12, 2, 8, 2, 10, 4, 2, 21, 2, 2, 8, 3, 4, 2, 3, 19, 5, 2, 8, 2, 2, 6, 2, 31, 2, 9, 10
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OFFSET

1,1


COMMENTS

Carmichael (1922) conjectured that the number 1 never appears in this sequence. Sierpiński conjectured and Ford (1998) proved that all integers greater than 1 occur in the sequence. Erdős (1958) proved that if s >= 1 appears in the sequence then it appears infinitely often.  Nick Hobson, Nov 04 2006
A002202(n) occurs a(n) times in A007614.  Reinhard Zumkeller, Nov 22 2015


REFERENCES

E. Lucas, Théorie des Nombres, Blanchard 1958.


LINKS

T. D. Noe, Table of n, a(n) for n=1..10000
R. D. Carmichael, Note on Euler's totient function, Bull. Amer. Math. Soc. 28 (1922), pp. 109110.
P. Erdős, Some remarks on Euler's totient function, Acta Arith. 4 (1958), pp. 1019.
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 2734.
N. Hobson, Problem 152, "Totient valence"
Eric Weisstein's World of Mathematics, Totient Valence Function


MATHEMATICA

max = 300; inversePhi[_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m * Times @@ (p/(p1)); n = m; nn = Reap[While[n <= nmax, If[EulerPhi[n] == m, Sow[n]]; n++]] // Last; If[nn == {}, {}, First[nn] ] ]; Reap[For[n = 1, n <= max, n = If[n == 1, 2, n+2], nn = inversePhi[n] ; If[nn != {} , Sow[nn // Length] ] ] ] // Last // First (* JeanFrançois Alcover, Nov 21 2013 *)


PROG

(Haskell)
import Data.List (group)
a058277 n = a058277_list !! (n1)
a058277_list = map length $ group a007614_list
 Reinhard Zumkeller, Nov 22 2015


CROSSREFS

The nonzero terms of A014197. Cf. A000010, A002202.
Cf. A007614.
Sequence in context: A211509 A305594 A320778 * A065852 A303998 A319712
Adjacent sequences: A058274 A058275 A058276 * A058278 A058279 A058280


KEYWORD

nonn,easy


AUTHOR

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001


EXTENSIONS

More terms from Nick Hobson, Nov 04 2006


STATUS

approved



