A007374 Smallest k such that phi(x) = k has exactly n solutions, n>=2.
1, 2, 4, 8, 12, 32, 36, 40, 24, 48, 160, 396, 2268, 704, 312, 72, 336, 216, 936, 144, 624, 1056, 1760, 360, 2560, 384, 288, 1320, 3696, 240, 768, 9000, 432, 7128, 4200, 480, 576, 1296, 1200, 15936, 3312, 3072, 3240, 864, 3120, 7344, 3888, 720, 1680, 4992
Offset: 2
References
- 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.
- Wacław Sierpiński, Elementary Theory of Numbers, p. 234, Warsaw, 1964.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Jud McCranie, Table of n, a(n) for n = 2..10000 (terms up to 1..1023 from T. D. Noe, terms 1024..7448 from Donovan Johnson)
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- Kevin Ford, The distribution of totients, Ramanujan J., (2) No. 1-2 (1998); New version of the 1998 article, arXiv:1104.3264 [math.NT], 2011-2013.
- Kevin Ford, The number of solutions of phi(x) = m, Annals of Mathematics 150:1 (1999), pp. 283-311.
- S. D. Merow, Has Carmichael's Totient Conjecture Been Proven? No, No, It Has Not, Notices Amer. Math. Soc., 66 (No. 5, 2019), 759-761.
- A. Schlafly and S. Wagon, Carmichael's conjecture on the Euler function is valid below 10^{10,000,000}, Mathematics of Computation, 63 No. 207 (1994), 415-419. See Table 2.
- Eric Weisstein's World of Mathematics, Phi function.
- Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture.
Crossrefs
Programs
-
Mathematica
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} ] -
PARI
v=vectorsmall(10^6);for(n=1,1e7,t=eulerphi(n);if(t<=#v,v[t]++)) u=vector(100);for(i=1,#v,t=v[i];if(t&&t<=#u&&u[t]==0,u[t]=i)); u[2..#u] \\ Charles R Greathouse IV, Oct 13 2014
Comments