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A221284 Numbers n such that phi(m) = n^2 for some m. 5

%I

%S 1,2,4,6,8,10,12,14,16,18,20,24,26,28,30,32,36,40,42,44,48,50,52,54,

%T 56,60,64,66,68,70,72,74,80,84,88,90,94,96,100,104,108,110,112,114,

%U 116,120,124,126,128,130,132,134,136,140,144,146,148,150,156,160,162,168,170

%N Numbers n such that phi(m) = n^2 for some m.

%H Charles R Greathouse IV, <a href="/A221284/b221284.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 a(n) = sqrt(A221285(n)).

%F Pollack and Pomerance show that n (log n)^0.0063 << a(n) << n (log n)^3.

%t 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 Reap[For[k = 1, k <= 200, k = k + If[k==1, 1, 2], If[inversePhiSingle[k^2] > 0, Print[k]; Sow[k]]]][[2, 1]] (* _Jean-Fran├žois Alcover_, Dec 11 2018 *)

%o (PARI) is(n)=istotient(n^2)

%Y Cf. A002202, A221285.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_, Feb 05 2013

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Last modified June 14 09:06 EDT 2021. Contains 345018 sequences. (Running on oeis4.)