A040117 Primes congruent to 5 (mod 12). Also primes p such that x^4 = 9 has no solution mod p.

5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269, 281, 293, 317, 353, 389, 401, 449, 461, 509, 521, 557, 569, 593, 617, 641, 653, 677, 701, 761, 773, 797, 809, 821, 857, 881, 929, 941, 953, 977, 1013, 1049, 1061, 1097, 1109, 1181, 1193

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Odd primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005, corrected by M. F. Hasler, Jul 03 2025

Complement of A040116 relative to A000040. - Vincenzo Librandi, Sep 17 2012

Odd primes of the form a^2 + b^2 such that a^2 == b^2 (mod 3). - Thomas Ordowski and Charles R Greathouse IV, May 20 2015

Yasutoshi Kohmoto observes that nextprime(a(n)) is more frequently congruent to 3 (mod 4) than to 1. This bias can be explained by the possible prime constellations and gaps: To have the same residue mod 4 as a prime in the list, the next prime must be at a gap of 4 or 8 or 12..., but a gap of 4 is impossible because 12k + 5 + 4 is divisible by 3, and gaps >= 12 are very rare for small primes. To have the residue 3 (mod 4) the next prime can be at a gap of 2 or 6 with no a priori divisibility property. However, this bias tends to disappear as the primes (and average prime gaps) grow bigger: for primes < 10^5, the ratio is about 35% vs 65% (as the above simple explanation suggests), but considering primes up to 10^8 yields a ratio of about 40% vs 60%. It can be expected that the ratio asymptotically tends to 1:1. - M. F. Hasler, Sep 01 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A068227, A068228, A068229, A068231, A068232, A068233, A068234, A068235.

Equal to A243183 (primes of the form 2x^2+2xy+5y^2) except for the additional A243183(1) = 2 (and indexing of subsequent terms).

Programs

Magma

[p: p in PrimesUpTo(1200) | not exists{x : x in ResidueClassRing(p) | x^4 eq 9} ]; // Vincenzo Librandi, Sep 17 2012

Mathematica

Select[Prime/@Range[250], Mod[ #, 12]==5&]
ok[p_]:= Reduce[Mod[x^4 - 9, p] == 0, x, Integers] == False;Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 17 2012 *)

PARI

for(i=1,250, if(prime(i)%12==5, print(prime(i))))

Formula

a(n) ~ 4n log n. - Charles R Greathouse IV, May 20 2015

Extensions

More terms from Dean Hickerson, Feb 27 2002