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A007374 Smallest k such that phi(x) = k has exactly n solutions, n>=2.

%I A007374 M1093 #66 May 30 2025 23:56:34
%S A007374 1,2,4,8,12,32,36,40,24,48,160,396,2268,704,312,72,336,216,936,144,
%T A007374 624,1056,1760,360,2560,384,288,1320,3696,240,768,9000,432,7128,4200,
%U A007374 480,576,1296,1200,15936,3312,3072,3240,864,3120,7344,3888,720,1680,4992
%N A007374 Smallest k such that phi(x) = k has exactly n solutions, n>=2.
%C A007374 The Carmichael Totient Conjecture is that there is no k such that phi(x) = k has a unique solution x. So a(1) does not exist.
%C A007374 Ford proved that a(n) exists for all n > 1. - _Charles R Greathouse IV_, Oct 13 2014
%D A007374 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
%D A007374 Wacław Sierpiński, Elementary Theory of Numbers, p. 234, Warsaw, 1964.
%D A007374 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007374 Jud McCranie, <a href="/A007374/b007374.txt">Table of n, a(n) for n = 2..10000</a> (terms up to 1..1023 from T. D. Noe, terms 1024..7448 from Donovan Johnson)
%H A007374 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A007374 Kevin Ford, <a href="https://doi.org/10.1023/A:1009761909132">The distribution of totients</a>, Ramanujan J., (2) No. 1-2 (1998); <a href="https://arxiv.org/abs/1104.3264">New version of the 1998 article</a>, arXiv:1104.3264 [math.NT], 2011-2013.
%H A007374 Kevin Ford, <a href="http://www.math.uiuc.edu/~ford/wwwpapers/sierp.pdf">The number of solutions of phi(x) = m</a>, Annals of Mathematics 150:1 (1999), pp. 283-311.
%H A007374 S. D. Merow, <a href="https://www.ams.org/journals/notices/201905/rnoti-p759.pdf">Has Carmichael's Totient Conjecture Been Proven? No, No, It Has Not</a>, Notices Amer. Math. Soc., 66 (No. 5, 2019), 759-761.
%H A007374 A. Schlafly and S. Wagon, <a href="https://doi.org/10.1090/S0025-5718-1994-1226815-3">Carmichael's conjecture on the Euler function is valid below 10^{10,000,000}</a>, Mathematics of Computation, 63 No. 207 (1994), 415-419. See Table 2.
%H A007374 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientValenceFunction.html">Phi function.</a>
%H A007374 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function Conjecture</a>.
%t A007374 a = Table[ 0, {10^5} ]; Do[ s = EulerPhi[ n ]; If[ s < 100001, a[ [ s ] ]++ ], {n, 1, 10^6} ]; Do[ k = 1; While[ a[ [ k ] ] != n, k++ ]; Print[ k ], {n, 2, 75} ]
%o A007374 (PARI) v=vectorsmall(10^6);for(n=1,1e7,t=eulerphi(n);if(t<=#v,v[t]++))
%o A007374 u=vector(100);for(i=1,#v,t=v[i];if(t&&t<=#u&&u[t]==0,u[t]=i)); u[2..#u]
%o A007374 \\ _Charles R Greathouse IV_, Oct 13 2014
%Y A007374 Essentially same as A014573. Records in A105207, A105208.
%Y A007374 Cf. A000010, A097942, A105207, A105208.
%K A007374 nonn,easy,nice
%O A007374 2,2
%A A007374 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_