A114345 Sequence of primes based on the powers of the golden mean; see formula section for description.

2, 3, 7, 11, 17, 47, 61, 97, 173, 367, 1367, 10631, 13781, 15919, 1008001, 2584403, 4232351, 5459719, 334525987, 11779122851, 13808301271, 116757956759, 2968189088940281, 32797072183910341, 5972846330691787903, 283950392369947419799, 2969782506626449546163

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..100

Formula

Let p(n) = floor(phi^(n/log(n))), g(n) = p(n) if (p(n) mod 2) = 0, otherwise g(n) = 0, and f(n) = f(n-1) + g(n) with f(1) = 2, f(2) = 3. Define b(n) as f(n) if f(n) is prime, then a(n) is the list of b(n) with duplicates removed.

Mathematica

p[n_]:= Floor[GoldenRatio^(n/Log[n])];
g[n_]:= g[n]= If[Mod[p[n], 2]==0, p[n], 0];
f[n_]:= f[n]= If[n<3, n+1, f[n-1] +g[n]];
DeleteDuplicates[Table[If[PrimeQ[f[n]], f[n], {}], {n, 1000}]]//Flatten

Prog

(SageMath)
def p(n): return floor(golden_ratio^(n/log(n)))
def g(n): return p(n) if (p(n)%2)==0 else 0
@CachedFunction
def f(n): return n+1 if (n<3) else f(n-1) + g(n)
def b(n): return f(n) if is_prime(f(n)) else {}
set([f(n) for n in range(1, 1001) if is_prime(f(n))]) # G. C. Greubel, Aug 09 2023

Crossrefs

Sequence in context: A350839 A161921 A060341 * A077165 A090666 A140409

Adjacent sequences: A114342 A114343 A114344 * A114346 A114347 A114348

Keyword

nonn,less

Author

Roger L. Bagula, Feb 08 2006

Extensions

Edited by G. C. Greubel, Aug 09 2023

Status

approved