A240920 Prime numbers that occur as divisors of numbers of the form m^2 + 5.

2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 127, 149, 163, 167, 181, 223, 227, 229, 241, 263, 269, 281, 283, 307, 347, 349, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487, 503, 509, 521, 523, 541, 547, 563

Offset: 1

Author

J. Lowell, Aug 02 2014

Keywords

• nonn
• easy

Comments

Conjecture: a prime number is in this sequence if and only if its next-to-last digit is even.

The law of quadratic reciprocity shows an odd prime is in the sequence if and only if it is 1, 3, 5, 7 or 9 (mod 20). This proves the above conjecture, so the sequence is the union of {2, 5} and A139513. - Jens Kruse Andersen, Aug 09 2014

Examples

			23 is in the sequence because it divides 8^2+5=69 with m=8.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
• David Lowry-Duda, Unexpected Conjectures about -5 Modulo Primes, College Mathematics Journal, Vol. 46, No. 1 (Jan 2015), pp.56-57.

Crossrefs

Cf. A002313 (k=1 or k=4), A033203 (k=2), A045331 (k=3), A139513.

Programs

Maple

isA240920 := proc(p)
    local n;
    if isprime(p) then
        for n from 0 to p do
            if modp(n^2+5,p) = 0 then
                return true;
            end if;
        end do:
        false;
    else
        false;
    end if;
end proc:
for i from 1 to 600 do
    p := ithprime(i) ;
    if isA240920(p) then
        printf("%d,",p);
    end if;
end do:

PARI

select(p->issquare(Mod(-5,p)), primes(100)) \\ Charles R Greathouse IV, Nov 29 2016

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016