A159828 a(n) is smallest number m > 0 such that m^2 + n^2 + 1 is prime.

1, 6, 1, 6, 9, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1, 6, 27, 8, 9, 24, 1, 6, 21, 4, 69, 12, 3, 6, 21, 6, 3, 6, 1, 6, 9, 2, 9, 6, 1, 6, 15, 6, 9, 6, 1, 6, 27, 2, 3, 36, 9, 6, 3, 6, 15, 18, 1, 48, 3, 4, 9, 6, 7, 6, 15, 4, 21, 42, 5, 6, 3, 2, 69, 18, 5, 6, 3, 2, 9, 24, 1, 6, 3, 8, 9, 6, 11, 18, 15, 4, 3, 6, 7, 18

Offset

1,2

Comments

a(2k-1) is odd, a(2k) is even.

There are infinitely many primes of the forms n^2 + m^2 and n^2 + m^2 + 1, but it is not known if the number of primes of the form n^2 + 1 is infinite; cf. comments in A002496, A002313, A079544.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Example

n = 1: 1^2 + 1^2 + 1 = 3 is prime, so a(1) = 1.
n = 2: 1^2 + 2^2 + 1 = 6, 2^2 + 2^2 + 1 = 9, 3^2 + 2^2 + 1 = 14, 4^2 + 2^2 + 1 = 21, 5^2 + 2^2 + 1 = 30 are composite, but 6^2 + 2^2 + 1 = 41 is prime, so a(2) = 6.
n = 27: 1^2 + 27^2 + 1 = 731 = 17*43, 2^2 + 27^2 + 1 = 734 = 2*367 are composite, but 3^2 + 27^2 + 1 = 739 is prime, so a(27) = 3.

Mathematica

snm[n_]:=Module[{c=n^2+1, x=NextPrime[n^2+1]}, While[!IntegerQ[Sqrt[x-c]], x= NextPrime[x]]; Sqrt[x-c]]; Array[snm, 100] (* Harvey P. Dale, Sep 22 2018 *)

Prog

(Magma) S:=[]; for n in [1..100] do q:=n^2+1; m:=1; while not IsPrime(m^2+q) do m+:=1; end while; Append(~S, m); end for; S; // Klaus Brockhaus, May 21 2009

Crossrefs

Cf. A069003 (smallest d such that n^2+d^2 is prime), A002496 (primes of form n^2+1), A002313 (primes of form x^2+y^2), A079544 (primes of form x^2+y^2+1, x>0, y>0).

Sequence in context: A222068 A272055 A157292 * A131114 A199230 A199101

Adjacent sequences: A159825 A159826 A159827 * A159829 A159830 A159831

Keyword

easy,nonn

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 23 2009

Extensions

Edited and extended by Klaus Brockhaus, May 21 2009

Status

approved