A125257 Smallest prime divisor of 4n^2+3 that is of the form 6k+1.

7, 19, 13, 67, 103, 7, 199, 7, 109, 13, 487, 193, 7, 787, 7, 13, 19, 433, 1447, 7, 19, 7, 13, 769, 2503, 2707, 7, 43, 7, 1201, 3847, 4099, 1453, 7, 4903, 7, 5479, 5779, 2029, 19, 7, 13, 7, 61, 37, 8467, 8839, 7, 13, 7, 3469, 31, 11239, 3889, 7, 12547, 7, 43, 19, 4801

Offset: 1

Author

Nick Hobson, Nov 26 2006

Keywords

• easy
• nonn

Comments

Any prime divisor of 4n^2+3 different from 3 is congruent to 1 modulo 6.

4n^2+3 is never a power of 3 for n > 0; hence a prime divisor congruent to 1 modulo 6 always exists.

a(n) = 7 if and only if n is congruent to 1 or -1 modulo 7.

Examples

			The prime divisors of 4*3^2+3=39 are 3 and 13, so a(3) = 13.
		

References

• D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.

Links

• Nick Hobson, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A057204, A124988.

Programs

Mathematica

Table[SelectFirst[FactorInteger[4 n^2+3][[;;,1]],Mod[#,6]==1&],{n,60}] (* Harvey P. Dale, Jan 17 2025 *)

PARI

vector(60, n, factor(4*n^2+3)[2-(n^2)%3,1])