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%I #48 Feb 13 2024 12:45:21
%S 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,
%T 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,
%U 2,10,4,2,21,2,2,8,3,4,2,3,19,5,2,8,2,2,6,2,31,2,9,10
%N Number of values of k such that phi(k) = n, where n runs through the values (A002202) taken by phi.
%C 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
%C A002202(n) occurs a(n) times in A007614. - _Reinhard Zumkeller_, Nov 22 2015
%D E. Lucas, Théorie des Nombres, Blanchard 1958.
%H T. D. Noe, <a href="/A058277/b058277.txt">Table of n, a(n) for n=1..10000</a>
%H R. D. Carmichael, <a href="http://dx.doi.org/10.1090/S0002-9904-1922-03504-5">Note on Euler's totient function</a>, Bull. Amer. Math. Soc. 28 (1922), pp. 109-110.
%H P. Erdős, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa413.pdf">Some remarks on Euler's totient function</a>, Acta Arith. 4 (1958), pp. 10-19.
%H M. Farrokhi D. G., <a href="/A058277/a058277.txt">Gap function to compute the inverse of Euler's totient function</a>
%H K. Ford, <a href="http://dx.doi.org/10.1090/S1079-6762-98-00043-2">The distribution of totients</a>, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
%H Nick Hobson, <a href="https://web.archive.org/web/20160328125112/http://qbyte.org/puzzles/p152s.html">Solution to puzzle 152: Totient valence</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>.
%t 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 *)
%o (Haskell)
%o import Data.List (group)
%o a058277 n = a058277_list !! (n-1)
%o a058277_list = map length $ group a007614_list
%o -- _Reinhard Zumkeller_, Nov 22 2015
%Y The nonzero terms of A014197. Cf. A000010, A002202.
%Y Cf. A007614.
%Y Cf. A006511 (largest k for which A000010(k) = A002202(n)).
%K nonn,easy
%O 1,1
%A Claude Lenormand (claude.lenormand(AT)free.fr), Jan 05 2001
%E More terms from _Nick Hobson_, Nov 04 2006
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