A218563 Numbers n such that n^2 + 1 is divisible by a 4th power.

182, 239, 443, 807, 1068, 1432, 1693, 2057, 2318, 2682, 2943, 3307, 3568, 3932, 4193, 4557, 4818, 5182, 5443, 5807, 6068, 6432, 6693, 7057, 7318, 7682, 7943, 8307, 8568, 8932, 9193, 9557, 9818, 10182, 10443, 10807, 11068, 11432, 11693, 12057, 12318, 12682

Offset: 1

Author

Michel Lagneau, Nov 02 2012

Keywords

• nonn

Comments

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

Examples

			239 is in the sequence because 239^2+1 = 57122 = 2*13^4;
27493 is in the sequence because 27493^2+1 = 755865050 = 2*5^2*17^4*181.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002522, A049532, A034939, A218562.

Programs

Maple

N:= 100000: # to get all terms <= N
res:= {}:
p:= 2;
while p^4 <= N^2+1 do
  for v in map(t -> subs(t,n), [msolve(n^2+1, p^4)]) do
    res:= res union {seq(k*p^4+v, k = 0 .. (N-v)/p^4)}
  od;
  p:= nextprime(p);
od:
sort(convert(res,list)); # Robert Israel, Oct 06 2016

Mathematica

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