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A221285 Square values taken by totient function phi(m) = A000010(m).

%I A221285 #17 Apr 29 2020 13:58:46
%S A221285 1,4,16,36,64,100,144,196,256,324,400,576,676,784,900,1024,1296,1600,
%T A221285 1764,1936,2304,2500,2704,2916,3136,3600,4096,4356,4624,4900,5184,
%U A221285 5476,6400,7056,7744,8100,8836,9216,10000,10816,11664,12100,12544,12996,13456,14400,15376,15876
%N A221285 Square values taken by totient function phi(m) = A000010(m).
%H A221285 Charles R Greathouse IV, <a href="/A221285/b221285.txt">Table of n, a(n) for n = 1..10000</a>
%H A221285 W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski, <a href="http://www.math.missouri.edu/~bbanks/papers/2004_mult_struct_Euler_function.pdf">Multiplicative structure of values of the Euler function</a>, in High primes and misdemeanours: Lectures in honour of the sixtieth birthday of Hugh Cowie Williams, Fields Inst. Comm. 41 (2004), pp. 29-47.
%H A221285 Tristan Freiberg, Carl Pomerance, <a href="http://arxiv.org/abs/1410.8109">A note on square totients</a>, arXiv:1410.8109 [math.NT], 2014.
%H A221285 Paul Pollack and Carl Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/squaretotients5.pdf">Square values of Euler's function</a>, Bulletin of the London Mathematical Society 46:2 (April 2014), pp. 403-414.
%F A221285 A002202 INTERSECTION A000290.
%F A221285 a(n) = A221284(n)^2.
%F A221285 Pollack & Pomerance show that n^2 log^.0126 n << a(n) << n^2 log^6 n.
%t A221285 inversePhiSingle[(m_)?EvenQ] := Module[{p, nmax, n}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; While[n <= nmax, If[EulerPhi[n] == m, Return[n]]; n++]; 0];
%t A221285 Reap[For[k = 1, k <= 200, k = k + If[k==1, 1, 2], If[inversePhiSingle[k^2] > 0, Print[k^2]; Sow[k^2]]]][[2, 1]] (* _Jean-François Alcover_, Dec 11 2018 *)
%o A221285 (PARI) is(n)=issquare(n) && istotient(n)
%Y A221285 Cf. A002202, A221284.
%K A221285 nonn
%O A221285 1,2
%A A221285 _Charles R Greathouse IV_, Feb 05 2013