A280988 Least k such that phi(k*n) is a perfect square, or 0 if no such k exists.

1, 1, 4, 2, 1, 2, 9, 1, 7, 1, 41, 1, 21, 9, 4, 2, 1, 6, 3, 2, 3, 41, 89, 2, 5, 14, 4, 13, 113, 2, 143, 1, 25, 1, 9, 3, 1, 2, 7, 1, 11, 3, 49, 25, 7, 89, 1151, 1, 43, 5, 4, 7, 553, 2, 15, 9, 1, 113, 233, 1, 77, 122, 1, 2, 21, 25, 299, 2, 356, 9, 281, 6, 3, 1, 11, 1, 61, 6, 313

Offset

1,3

Comments

Pollack and Pomerance proved that if phi(a) = b^m, then m = 2 occurs only on a set of density 0.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Paul Pollack and Carl Pomerance, Square values of Euler's function, Bulletin of the London Mathematical Society, Vol. 46, No. 2 (2014), pp. 403-414, alternative link.

Example

a(11) = 41 because phi(k*11) is not a perfect square for 0 < k < 41 and phi(41*11) = 20^2.

Maple

f:= proc(n) local k;
    for k from 1 do
      if issqr(numtheory:-phi(k*n)) then return k fi
   od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 12 2017

Mathematica

a[n_] := Module[{k = 1}, While[!IntegerQ[Sqrt[EulerPhi[k*n]]], k++]; k]; Array[a, 80] (* Amiram Eldar, Jul 13 2019 *)

Prog

(PARI) a(n) = {my(k = 1); while (!issquare(eulerphi(k*n)), k++); k; }

Crossrefs

Cf. A000010, A039770, A062732, A280986.

Sequence in context: A229974 A364789 A281065 * A175665 A200586 A097525

Adjacent sequences: A280985 A280986 A280987 * A280989 A280990 A280991

Keyword

nonn

Author

Altug Alkan, Jan 12 2017

Status

approved