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Numbers k such that cototient(k) is a square.
12

%I #111 Sep 08 2022 08:45:04

%S 1,2,3,5,6,7,8,11,13,17,19,21,23,24,27,28,29,31,32,37,41,43,47,53,54,

%T 59,61,67,68,69,71,73,79,83,89,96,97,101,103,107,109,112,113,124,125,

%U 127,128,131,133,137,139,141,149,151,157,163,167,173,179,181,189,191

%N Numbers k such that cototient(k) is a square.

%C Some different families and subsequences of integers belong to this sequence, see the file "Subfamilies and subsequences" for more details, with data, comments, proofs, formulas and examples. - _Bernard Schott_, Mar 05 2019

%H Harry J. Smith, <a href="/A063752/b063752.txt">Table of n, a(n) for n = 1..1000</a>

%H Thomas E. Moore, <a href="https://cms.math.ca/crux/backfile/Crux_v14n03_Mar.pdf">Problem 1204</a>, Crux Mathematicorum, page 93, Vol. 14, Mar. 88.

%H Bernard Schott, <a href="/A063752/a063752_7.pdf">Subfamilies and subsequences</a>

%F a(n) seems to be asymptotic to c * n * log(n) with c = 1.7... (all primes are in the sequence since cototient(p) = 1). - _Benoit Cloitre_, Sep 08 2002

%t Select[Range[200], IntegerQ[Sqrt[# - EulerPhi[#]]]&] (* _Jean-François Alcover_, Nov 06 2016 *)

%o (PARI) j=[]; for(n=1,400,x=n-eulerphi(n); if(issquare(x),j=concat(j,n))); j

%o (PARI) { n=0; for (m=1, 10^9, if (issquare(m - eulerphi(m)), write("b063752.txt", n++, " ", m); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 29 2009

%o (Magma) [n: n in [1..200] | IsSquare(n - EulerPhi(n))]; // _Vincenzo Librandi_, Jan 11 2019

%Y Cf. A000010, A051953.

%Y Subsequences: A000396, A246551, A323916, A323917, A323918, A306670.

%K nonn

%O 1,2

%A _Jason Earls_, Aug 11 2001