A181413 a(n) is the smallest number such that a(n)^2 + 1 is divisible by A002144(1)* A002144(2)*...* A002144(n).

2, 8, 47, 2163, 18543, 241727, 3101272, 842894268, 8245041748, 521781374353, 101476250977928, 671795954794788, 32126984574675193, 425090834074746637, 309609468228403885693, 25836182225971546313682, 38544366727563360743217, 217758730168965028986551783, 25789605237863389220212237968, 309600287787935978580674202007

Offset: 1

Author

Michel Lagneau, Jan 28 2011

Keywords

• nonn

Examples

			a(1) = 2 because A002144(1) | 2^2+1 = 5 ;
a(2)=8 because A002144(1) * A002144(2) | 8^2+1 = 5*13 ;
a(6) = 241727 because A002144(1) * A002144(2)*...* A002144(6) | 241727^2+1
  = 2 * 5 * 13 * 17 * 29 * 37 * 41 * 601.
		

Crossrefs

Cf. A002144 (Pythagorean primes: primes of form 4n+1) A002731.

Programs

Maple

with(numtheory):nn:=1000:T:=array(1..1000):k:=1:for x from 1 to nn do: p:=4*x+1:if
  type(p, prime)=true then T[k]:=p:k:=k+1:else fi:od:pr:=1:for n from 1 to k do:
  pp:=pr*T[n] :ind:=0:for q from 1 to pp while (ind=0) do: z:=q^2+1:if irem(z,pp)=0
  and ind = 0 then ind: = 1:pr:=pp:print( q):else fi:od:od:
# Alternative
PP:= select(isprime, [seq(i,i=5..200,4)]):
f:= n -> min(map(t -> rhs(op(t)),[msolve(x^2+1, convert(PP[1..n],`*`))])):
map(f, [$1..20]); # Robert Israel, Feb 01 2019

Extensions

More terms from Robert Israel, Feb 01 2019