A047650 Primes for which golden mean tau is a quadratic residue.

29, 89, 101, 181, 229, 349, 401, 461, 509, 521, 541, 709, 761, 769, 809, 941, 1009, 1021, 1049, 1061, 1109, 1229, 1249, 1289, 1361, 1409, 1549, 1601, 1621, 1669, 1709, 1721, 1741, 1789, 1861, 2029, 2069, 2081, 2089, 2389, 2441, 2621, 2729, 2801, 2861

Offset

0,1

Comments

Primes of the form x^2 + 20*y^2. - T. D. Noe, May 08 2005

Also primes p that divide the sum of cubes of the first (p-1)/2 Fibonacci numbers A005968((p-1)/2). - Alexander Adamchuk, Aug 07 2006

From A.H.M. Smeets, Nov 16 2023: (Start)

Mean gap size between two consecutive terms at p: ~ 8*log(p).

In x^2 + 20y^2: x == 1 (mod 2) and x !== 5 (mod 10). Otherwise not prime. (End)

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..20000 (terms 0..1700 from Vincenzo Librandi)

Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.

E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138.

E. Lehmer, Correction, Fib. Quart., 4 (1966), 135-138.

E. Lehmer, On the quadratic character of the Fibonacci root (annotated scanned copy)

Formula

From A.H.M. Smeets, Nov 16 2023: (Start)

Equals {prime(n): A296240(n) in {2^k: k > 0}} = {A308787} union {A308789} union {A308793} union ... .

a(n) ~ A000040(8*n). (End)

Mathematica

nn=20; pMax=3000; Union[Reap[Do[p=x^2 + nn*y^2; If[p<=pMax&&PrimeQ[p], Sow[p]], {x, Sqrt[pMax]}, {y, Sqrt[pMax/nn]}]][[2, 1]]] (* Vincenzo Librandi, Sep 05 2016 *)

Prog

(Magma) k:=20; [p: p in PrimesUpTo(3000) | NormEquation(k, p) eq true]; // Vincenzo Librandi, Sep 05 2016

Crossrefs

Cf. A047651, A001583.

Cf. A005968.

Sequence in context: A042656 A042658 A042660 * A308787 A141883 A142791

Adjacent sequences: A047647 A047648 A047649 * A047651 A047652 A047653

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Jan 25 2000

Status

approved