

A007367


Numbers k such that phi(x) = k has exactly 3 solutions.
(Formerly M2163)


14



2, 44, 56, 92, 104, 116, 140, 164, 204, 212, 260, 296, 332, 344, 356, 380, 392, 444, 452, 476, 524, 536, 564, 584, 588, 620, 632, 684, 692, 716, 744, 764, 776, 836, 860, 884, 932, 956, 980, 1004, 1016, 1112, 1124, 1136, 1172, 1196, 1284, 1292, 1304
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OFFSET

1,1


COMMENTS

For known terms:
 The greatest common divisor of the three solutions is the distance of the middle solution from the least solution and is half the distance of the middle solution to the largest solution.
 If the number of distinct prime factors of k equals the number of solutions of k = phi(x), then the greatest common divisor of the solutions is the least solution divided by the number of solutions.
 Except for a(1), if the largest prime factor is the same for all solutions and is equal to the greatest common divisor of all solutions then the distance from a(n) to the least solution is gcd({k: phi(k) = a(n)}) + 2. (End)
By Ford's theorem on Euler totient function, this sequence is infinite.  Jianing Song, Jul 18 2018


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.
J.M. De Koninck, Ces nombres qui nous fascinent, Entry 44, p. 17, Ellipses, Paris 2008.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

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].


EXAMPLE

phi(69) = phi(92) = phi(138) = 44, so 44 is a term.


MATHEMATICA

a = Table[ 0, {1500} ]; Do[ p = EulerPhi[ n ]; If[ p < 1501, a[ [ p ] ]++ ], {n, 1, 1500} ]; Select[ Range[ 1500 ], a[ [ # ] ] == 3 & ]
Take[Select[Tally[EulerPhi[Range[50000]]], #[[2]]==3&][[All, 1]], 50]//Sort (* Harvey P. Dale, Apr 02 2018 *)


PROG

(Haskell)
a007367 n = a007367_list !! (n1)
a007367_list = map fst $
filter ((== 3) . snd) $ zip a002202_list a058277_list


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



