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%I M4685 #105 Nov 08 2022 20:41:41
%S 1,10,22,28,30,46,52,54,58,66,70,78,82,102,106,110,126,130,136,138,
%T 148,150,166,172,178,190,196,198,210,222,226,228,238,250,262,268,270,
%U 282,292,294,306,310,316,330,342,346,358,366,372,378,382,388,418,430,438
%N Numbers k such that phi(x) = k has exactly 2 solutions.
%C Contains {2*3^(6k+1): k >= 1} as a subsequence. This is the simplest proof for the infinity of these numbers (see Sierpiński, Exercise 12, p. 237). - _Franz Vrabec_, Aug 21 2021
%C The smaller of the solutions to phi(x) = a(n) is given by A271983(n). It is conjectured that the larger solution is 2*A271983(n); or equivalently, all terms in A271983 are odd. - _Jianing Song_, Nov 08 2022
%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 W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007366/b007366.txt">Table of n, a(n) for n = 1..1000</a>
%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 R. G. Wilson v, <a href="/A007015/a007015.pdf">Letter to N. J. A. Sloane, Jul. 1992</a>
%F #({phi^-1(a(n))}) = 2. - _Torlach Rush_, Dec 22 2017
%e 10 = phi(11) = phi(22).
%p select(nops@numtheory:-invphi=2, [$1..1000]); # _Robert Israel_, Dec 20 2017
%t a = Table[ 0, {500} ]; Do[ p = EulerPhi[ n ]; If[ p < 501, a[ [ p ] ]++ ], {n, 1, 500} ]; Select[ Range[ 500 ], a[ [ # ] ] == 2 & ]
%t (* Second program: *)
%t With[{nn = 1325}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], Last@ # == 2 &][[All, 1]], # < nn/3 &] ] (* _Michael De Vlieger_, Dec 20 2017 *)
%Y Cf. A000010, A001221, A007367, A023900, A271983.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_
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