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

1,1

Comments

Primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005.

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

Formula

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

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  *)

Prog

(PARI) for(i=1, 250, if(prime(i)%12==5, print(prime(i))))
(Magma) [p: p in PrimesUpTo(1200) | not exists{x : x in ResidueClassRing(p) | x^4 eq 9} ]; // Vincenzo Librandi, Sep 17 2012

Crossrefs

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

Apparently a subsequence of A243183. - N. J. A. Sloane, Jun 02 2014

Sequence in context: A030554 A017581 A068230 * A145471 A126193 A074965

Adjacent sequences: A040114 A040115 A040116 * A040118 A040119 A040120

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Dean Hickerson, Feb 27 2002

Status

approved