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A007374 Smallest k such that phi(x) = k has exactly n solutions.
(Formerly M1093)
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Carmichael conjectured that no term exists for n=1.

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

T. D. Noe and Donovan Johnson, Table of n, a(n) for n = 2..7448 (terms up to a(1023) from T. D. Noe)

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 number of solutions of phi(x) = m, Annals of Mathematics 150:1 (1999), pp. 283-311.

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.

Cf. A000010, A097942, A105207, A105208.

Sequence in context: A187941 A085083 A076745 * A105207 A202148 A215825

Adjacent sequences:  A007371 A007372 A007373 * A007375 A007376 A007377

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

EXTENSIONS

Link fixed by Charles R Greathouse IV, Oct 06 2009

STATUS

approved

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Last modified February 16 02:39 EST 2019. Contains 320140 sequences. (Running on oeis4.)