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URL: https://oeis.org/A007374

⇱ A007374 - OEIS

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A007374
Smallest k such that phi(x) = k has exactly n solutions, n>=2.
20
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
2,2
COMMENTS
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.
Ford proved that a(n) exists for all n > 1. - Charles R Greathouse IV, Oct 13 2014
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.
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} ]
PROG
(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
CROSSREFS
Essentially same as A014573. Records in A105207, A105208.
Sequence in context: A187941 A085083 A076745 * A105207 A202148 A215825
KEYWORD
nonn,easy,nice
STATUS
approved