A387306 Primes p == 3 (mod 4) such that neither 2p-1 nor 2p+3 is a sum of two squares.

107, 191, 359, 443, 587, 719, 751, 863, 919, 947, 983, 1091, 1223, 1283, 1367, 1439, 1451, 1619, 1667, 1699, 1783, 1871, 1907, 1951, 2039, 2207, 2339, 2447, 2459, 2687, 2699, 2711, 2803, 2879, 3023, 3271, 3359, 3371, 3467, 3491, 3623, 3719, 3851, 3863, 3923

Offset

1,1

Comments

These primes are the exceptional primes p == 3 (mod 4) that do not admit a representation of the form p = u*(u+1) + v*(v+1) +-1.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

For p = 47 == 3 (mod 4), we have 2p-1 = 93, which is not a sum of two squares. But 2p+3 = 97 = 9^2 + 4^2 is a sum of two squares, so 47 is not in this sequence.
For p = 107 == 3 (mod 4), we compute 2p-1 = 213 and 2p+3 = 217. Neither 213 nor 217 is a sum of two squares. Therefore 107 is in this sequence.

Maple

ns2:= proc(n)
   ormap(t -> t[1] mod 4 = 3 and t[2]::odd, ifactors(n)[2])
end proc:
filter:= n -> isprime(n) and ns2(2*n-1) and ns2(2*n+3):
select(filter, [seq(i, i=3..4000, 4)]); # Robert Israel, Sep 04 2025

Crossrefs

Cf. A002145 (primes == 3 (mod 4)), A001481 (sum of two squares, zeros allowed), A022544 (not a sum of two squares, zeros allowed).

Sequence in context: A142142 A265915 A210361 * A250147 A142270 A044339

Adjacent sequences: A387303 A387304 A387305 * A387307 A387308 A387309

Keyword

nonn

Author

Charles Kusniec, Aug 25 2025

Status

approved