OFFSET
1,1
COMMENTS
Primes of the form 2x^2-2xy+5y^2 with x and y nonnegative. - T. D. Noe, May 08 2005.
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Dean Hickerson, Feb 27 2002
STATUS
approved