A059791 Numbers n such that floor(phi^n) is prime, where phi = golden ratio.

2, 5, 6, 7, 11, 13, 17, 19, 24, 31, 37, 41, 47, 48, 53, 61, 71, 79, 96, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667

Offset: 1

Author

Naohiro Nomoto, Feb 22 2001

Keywords

• nonn

Comments

Tested up to n=250000. - Mark Rodenkirch, Feb 27 2020

Examples

			floor(phi^863)=227160876495918562748535035942584201965901433059749617\
427535706949917136103176482875403653972639455945062095866005032008819\
9236184776437699830957031191632116265394965429613743580479 is prime.
		

Links

• Eric Weisstein's World of Mathematics, Phi-Prime

Crossrefs

Cf. A001622, A059792.

Programs

Mathematica

Block[{$MaxExtraPrecision=10000}, Select[Range[14000],PrimeQ[ Floor[ GoldenRatio^#]]&]] (* Harvey P. Dale, Mar 06 2017 *)

PARI

isok(n) = isprime(floor(((sqrt(5)+1)/2)^n)) \\ Michel Marcus, Jul 14 2013
Terms generated and tested with pfgw then verified with PARI using the following:

PARI

c(n) = 3*fibonacci(n-1) + fibonacci(n-2) + (n % 2) - 1; ispseudoprime(c(n)) \\ Mark Rodenkirch, Feb 27 2020

Extensions

More terms from Vladeta Jovovic, Feb 24 2001

a(27)-a(33) from Eric W. Weisstein, May 01 2006

a(34)-a(36) from Dmitry Kamenetsky, Dec 29 2008

a(37)-a(52) from Mark Rodenkirch, Feb 27 2020