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A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

%I A293928 #86 Dec 04 2024 09:47:11
%S A293928 1,2,4,6,8,12,16,18,20,24,32,36,40,48,54,64,72,80,84,96,100,108,120,
%T A293928 128,144,160,162,168,192,200,216,240,252,256,272,288,312,320,324,336,
%U A293928 360,384,400,432,440,480,486,500,504,512,544,576,588,600,624,640,648,672,684
%N A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).
%C A293928 "Totients" are terms of A000010. - _N. J. A. Sloane_, Oct 22 2017
%C A293928 The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions to phi(m)^2 = phi(phi(m)*m).
%C A293928 The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...
%C A293928 Where (if ever) does this first differ from A068997? - _R. J. Mathar_, Oct 30 2017
%C A293928 Apparently the set of the m is A151999. - _R. J. Mathar_, Mar 25 2024
%C A293928 If m satisfies phi(m)^2 = phi(phi(m)*m), then it satisfies phi(m)^(k+1) = phi(phi(m)^k*m) for all k >= 1. - _Max Alekseyev_, Dec 03 2024
%H A293928 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%e A293928 96 is a term since 96^2 = phi(96*288), with m=288 where phi(288) = 96.
%o A293928 (PARI) isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)););); return (0);} \\ using the invphi script by Max Alekseyev; _Michel Marcus_, Nov 01 2017
%Y A293928 Cf. A000010, A006511, A032447, A007366.
%Y A293928 Subsequence of A002202.
%K A293928 nonn
%O A293928 1,2
%A A293928 _Torlach Rush_, Oct 19 2017
%E A293928 More terms from _Michel Marcus_, Oct 24 2017
%E A293928 Definition simplified by _Max Alekseyev_, Dec 03 2024