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

Author

Jason Earls, Aug 11 2001

Keywords

• nonn

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

Crossrefs

Cf. A000010, A051953.

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

Programs

Magma

[n: n in [1..200] | IsSquare(n - EulerPhi(n))]; // Vincenzo Librandi, Jan 11 2019

Mathematica

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

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

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