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Numbers k such that phi(x) = k has exactly 12 solutions.
11

%I #14 Nov 17 2024 07:20:25

%S 160,168,352,448,816,928,972,1024,1176,1848,2464,3040,3808,4152,4440,

%T 4512,4736,4944,5104,5152,5160,5304,5952,6408,6656,6672,6784,7648,

%U 8384,8704,8904,10432,10528,10624,11000,11008,11456,11776,12048,12416,13024,13032

%N Numbers k such that phi(x) = k has exactly 12 solutions.

%H Amiram Eldar, <a href="/A060675/b060675.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%e 160 = phi(187) = phi(205) = phi(328) = phi(352) = phi(374) = phi(400) = phi(410) = phi(440) = phi(492) = phi(528) = phi(600) = phi(660).

%t a = Table[ 0, {15000} ]; Do[ p = EulerPhi[ n ]; If[ p < 15001, a[ [ p ] ]++ ], {n, 1, 60000} ]; Select[ Range[ 15000 ], a[ [ # ] ] == 12 & ]

%t Union[Transpose[Select[Tally[EulerPhi[Range[100000]]],#[[2]]==12&]][[1]]] (* _Harvey P. Dale_, Oct 05 2015 *)

%o (PARI) is(n)=sum(i=1,n,eulerphi(i)==n)==12 \\ _Charles R Greathouse IV_, Mar 03 2014

%o (PARI) is(k) = invphiNum(k) == 12 \\ _Amiram Eldar_, Nov 17 2024, using _Max Alekseyev_'s invphi.gp

%Y Cf. A000010.

%Y Number of solutions: A007617 (0), A007366 (2), A007367 (3), A060667 (4), A060668 (5), A060669 (6), A060670 (7), A060671 (8), A060672 (9), A060673 (10), A060674 (11), this sequence (12).

%K nonn

%O 1,1

%A _Robert G. Wilson v_, Apr 18 2001