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%I #57 Mar 06 2023 12:56:52
%S 1,2,5,8,10,12,17,32,34,37,40,48,57,60,63,74,76,85,101,108,114,125,
%T 126,128,136,160,170,185,192,197,202,204,219,240,250,257,273,285,292,
%U 296,304,315,364,370,380,394,401,432,438,444,451,456,468,489,504,505
%N Numbers k such that phi(k) is a perfect square.
%C A004171 is a subsequence because phi(2^(2k+1)) = (2^k)^2. - _Enrique Pérez Herrero_, Aug 25 2011
%C Subsequence of primes is A002496 since in this case phi(k^2+1) = k^2. - _Bernard Schott_, Mar 06 2023
%D D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 141.
%H T. D. Noe, <a href="/A039770/b039770.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://math.dartmouth.edu/~carlp/PDF/banksfinal2.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 (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
%H P. Pollack and C. Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/squaretotients8.pdf">Square values of Euler's function</a>, submitted for publication.
%H Bernard Schott, <a href="/A039770/a039770.pdf">Subfamilies and subsequences</a>
%F a(n) seems to be asymptotic to c*n^(3/2) with 1 < c < 1.3. - _Benoit Cloitre_, Sep 08 2002
%F Banks, Friedlander, Pomerance, and Shparlinski show that a(n) = O(n^1.421). - _Charles R Greathouse IV_, Aug 24 2009
%e phi(34) = 16 = 4*4.
%p with(numtheory); isA039770 := proc (n) return issqr(phi(n)) end proc; seq(`if`(isA039770(n), n, NULL), n = 1 .. 505); # _Nathaniel Johnston_, Oct 09 2013
%t Select[ Range[ 600 ], IntegerQ[ Sqrt[ EulerPhi[ # ] ] ]& ]
%o (PARI) for(n=1, 120, if (issquare(eulerphi(n)), print1(n, ", ")))
%Y Cf. A000010, A007614. A062732 gives the squares. A306882 (squares not totient).
%Y Cf. A068560, A114063, A078164.
%Y Subsequences: A054755, A002496, A004171, A262406, A324745, A324746, A247129, A324747, A306908.
%K nonn,easy,nice
%O 1,2
%A _Olivier Gérard_
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