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

1, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 576, 676, 784, 900, 1024, 1296, 1600, 1764, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4096, 4356, 4624, 4900, 5184, 5476, 6400, 7056, 7744, 8100, 8836, 9216, 10000, 10816, 11664, 12100, 12544, 12996, 13456, 14400, 15376, 15876

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.
• Tristan Freiberg, Carl Pomerance, A note on square totients, arXiv:1410.8109 [math.NT], 2014.
• 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, A221284.

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^2]; Sow[k^2]]]][[2, 1]] (* Jean-François Alcover, Dec 11 2018 *)

PARI

is(n)=issquare(n) && istotient(n)

Formula

A002202 INTERSECTION A000290.

a(n) = A221284(n)^2.

Pollack & Pomerance show that n^2 log^.0126 n << a(n) << n^2 log^6 n.