A218563 - OEIS

%I #15 Oct 06 2016 02:44:16

%S 182,239,443,807,1068,1432,1693,2057,2318,2682,2943,3307,3568,3932,

%T 4193,4557,4818,5182,5443,5807,6068,6432,6693,7057,7318,7682,7943,

%U 8307,8568,8932,9193,9557,9818,10182,10443,10807,11068,11432,11693,12057,12318,12682

%N Numbers n such that n^2 + 1 is divisible by a 4th power.

%C Includes all n == 182 or 443 (mod 625). In particular, the sequence has positive asymptotic density. # _Robert Israel_, Oct 06 2016

%H Robert Israel, <a href="/A218563/b218563.txt">Table of n, a(n) for n = 1..10000</a>

%e 239 is in the sequence because 239^2+1 = 57122 = 2*13^4;

%e 27493 is in the sequence because 27493^2+1 = 755865050 = 2*5^2*17^4*181.

%p N:= 100000: # to get all terms <= N

%p res:= {}:

%p p:= 2;

%p while p^4 <= N^2+1 do

%p   for v in map(t -> subs(t,n), [msolve(n^2+1, p^4)]) do

%p     res:= res union {seq(k*p^4+v, k = 0 .. (N-v)/p^4)}

%p   od;

%p   p:= nextprime(p);

%p od:

%p sort(convert(res,list)); # _Robert Israel_, Oct 06 2016

%t Select[Range[2,13000],Max[Transpose[FactorInteger[#^2+1]][[2]]]>3&]

%Y Cf. A002522, A049532, A034939, A218562.

%K nonn

%O 1,1

%A _Michel Lagneau_, Nov 02 2012