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

1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 30, 32, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 74, 80, 84, 88, 90, 94, 96, 100, 104, 108, 110, 112, 114, 116, 120, 124, 126, 128, 130, 132, 134, 136, 140, 144, 146, 148, 150, 156, 160, 162, 168, 170

Offset: 1

Author

Charles R Greathouse IV, Feb 05 2013

Keywords

• nonn

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• W. D. Banks, J. B. Friedlander, C. Pomerance, and I. E. Shparlinski, Multiplicative structure of values of the Euler function, in High primes and misdemeanours: Lectures in honour of the sixtieth birthday of Hugh Cowie Williams, Fields Inst. Comm. 41 (2004), pp. 29-47.
• Paul Pollack and Carl Pomerance, Square values of Euler's function, Bulletin of the London Mathematical Society 46:2 (April 2014), pp. 403-414.

Crossrefs

Cf. A002202, A221285.

Programs

Mathematica

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];
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-François Alcover, Dec 11 2018 *)

PARI

is(n)=istotient(n^2)

Formula

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

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