A249077 Primes of the form n^2 + k such that n^2 - k is also prime, where -n < k < n.

3, 5, 7, 11, 13, 19, 31, 41, 61, 67, 73, 79, 83, 89, 97, 103, 137, 139, 149, 151, 157, 181, 193, 199, 211, 223, 227, 239, 241, 271, 311, 317, 331, 337, 349, 373, 421, 433, 439, 443, 449, 461, 607, 619, 631, 643, 661, 691, 719, 739, 757, 811, 823, 829, 853, 859

Offset: 1

Author

Arkadiusz Wesolowski, Oct 20 2014

Keywords

• nonn

Comments

Members of a pair (a, b) of primes such that a < b and the distances from a and b to the nearest square above a (or below b) are equal.

The only prime of the form n^2 + 1 (A002496) in the sequence is 5.

Is this sequence infinite?

Examples

			2^2-1=3, 2^2+1=5, both prime.
8^2-3=61, 8^2+3=67, both prime.
		

Crossrefs

Cf. A047972, A053187.

Programs

Magma

lst:=[]; for m in [1..28] do r:=m*(m+1)+1; s:=(m+1)^2; for a in [r..s-1] do if IsPrime(a) then b:=2*s-a; if IsPrime(b) then Append(~lst, a); Append(~lst, b); end if; end if; end for; end for; Sort(lst);

Maple

g:= proc(t,m) if isprime(m+t) and isprime(m-t) then (m+t,m-t) else NULL fi end proc:
`union`(seq(map(g,{$1..n-1},n^2),n=2..100));
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list));
# Robert Israel, Oct 31 2014

PARI

for(n=1, 859, if(issquare(n), x=ps=n; until(issquare(x), x++); ns=x); if(isprime(n), if(n-ps
				

Formula

A prime p is in the sequence if and only if 2*A053187(p)-p is prime.