A063752 Numbers k such that cototient(k) is a square.

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, 59, 61, 67, 68, 69, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 124, 125, 127, 128, 131, 133, 137, 139, 141, 149, 151, 157, 163, 167, 173, 179, 181, 189, 191

Offset

1,2

Comments

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

Links

Harry J. Smith, Table of n, a(n) for n = 1..1000

Thomas E. Moore, Problem 1204, Crux Mathematicorum, page 93, Vol. 14, Mar. 88.

Bernard Schott, Subfamilies and subsequences

Formula

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

Mathematica

Select[Range[200], IntegerQ[Sqrt[# - EulerPhi[#]]]&] (* Jean-François Alcover, Nov 06 2016 *)

Prog

(PARI) j=[]; for(n=1, 400, x=n-eulerphi(n); if(issquare(x), j=concat(j, n))); j
(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
(Magma) [n: n in [1..200] | IsSquare(n - EulerPhi(n))]; // Vincenzo Librandi, Jan 11 2019

Crossrefs

Cf. A000010, A051953.

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

Sequence in context: A123030 A284836 A267300 * A191893 A016741 A191167

Adjacent sequences: A063749 A063750 A063751 * A063753 A063754 A063755

Keyword

nonn

Author

Jason Earls, Aug 11 2001

Status

approved