A007374 - OEIS

%I M1093 #55 Feb 16 2025 08:32:31

%S 1,2,4,8,12,32,36,40,24,48,160,396,2268,704,312,72,336,216,936,144,

%T 624,1056,1760,360,2560,384,288,1320,3696,240,768,9000,432,7128,4200,

%U 480,576,1296,1200,15936,3312,3072,3240,864,3120,7344,3888,720,1680,4992

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

%C 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 Ford proved that a(n) exists for all n > 1. - _Charles R Greathouse IV_, Oct 13 2014

%D 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 Wacław Sierpiński, Elementary Theory of Numbers, p. 234, Warsaw, 1964.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Donovan Johnson, <a href="/A007374/b007374.txt">Table of n, a(n) for n = 2..7448</a> (terms up to a(1023) from T. D. Noe)

%H 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 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientValenceFunction.html">Phi function.</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function Conjecture</a>.

%t 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 (PARI) v=vectorsmall(10^6);for(n=1,1e7,t=eulerphi(n);if(t<=#v,v[t]++))

%o u=vector(100);for(i=1,#v,t=v[i];if(t&&t<=#u&&u[t]==0,u[t]=i)); u[2..#u]

%o \\ _Charles R Greathouse IV_, Oct 13 2014

%Y Essentially same as A014573. Records in A105207, A105208.

%Y Cf. A000010, A097942, A105207, A105208.

%K nonn,easy,nice

%O 2,2

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_