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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 (list; graph; refs; listen; history; text; internal format)
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. 109-110.

P. Erdős, Some remarks on Euler's totient function, Acta Arith. 4 (1958), pp. 10-19.

K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.

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/(p-1)); 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 (* Jean-François Alcover, Nov 21 2013 *)

PROG

(Haskell)

import Data.List (group)

a058277 n = a058277_list !! (n-1)

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: A076410 A211509 A305594 * 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

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Last modified October 21 20:29 EDT 2018. Contains 316428 sequences. (Running on oeis4.)