cp's OEIS Frontend

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

%I A280988 #27 Jul 13 2019 09:17:21
%S A280988 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,
%T A280988 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,
%U A280988 113,233,1,77,122,1,2,21,25,299,2,356,9,281,6,3,1,11,1,61,6,313
%N A280988 Least k such that phi(k*n) is a perfect square, or 0 if no such k exists.
%C A280988 Pollack and Pomerance proved that if phi(a) = b^m, then m = 2 occurs only on a set of density 0.
%H A280988 Amiram Eldar, <a href="/A280988/b280988.txt">Table of n, a(n) for n = 1..10000</a>
%H A280988 Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1112/blms/bdt097">Square values of Euler's function</a>, Bulletin of the London Mathematical Society, Vol. 46, No. 2 (2014), pp. 403-414, <a href="https://math.dartmouth.edu/~carlp/squaretotients10.pdf">alternative link</a>.
%e A280988 a(11) = 41 because phi(k*11) is not a perfect square for 0 < k < 41 and phi(41*11) = 20^2.
%p A280988 f:= proc(n) local k;
%p A280988     for k from 1 do
%p A280988       if issqr(numtheory:-phi(k*n)) then return k fi
%p A280988    od
%p A280988 end proc:
%p A280988 map(f, [$1..100]); # _Robert Israel_, Jan 12 2017
%t A280988 a[n_] := Module[{k = 1}, While[!IntegerQ[Sqrt[EulerPhi[k*n]]], k++]; k]; Array[a, 80] (* _Amiram Eldar_, Jul 13 2019 *)
%o A280988 (PARI) a(n) = {my(k = 1); while (!issquare(eulerphi(k*n)), k++); k; }
%Y A280988 Cf. A000010, A039770, A062732, A280986.
%K A280988 nonn
%O A280988 1,3
%A A280988 _Altug Alkan_, Jan 12 2017