A265481 Numbers k such that 9^k - 3^k - 1 is prime.

1, 2, 3, 6, 7, 20, 35, 36, 140, 523, 1170, 1731, 1842, 3727, 3886, 9270, 11071, 13823, 14451, 27086, 27606, 31876, 78008

Offset: 1

Author

Vincenzo Librandi, Dec 09 2015

Keywords

• nonn
• more

Comments

For k = 1, 2, 3, 6, 7, 20, 35, the corresponding primes are 5, 71, 701, 530711, 4780781, 12157665455570144399, 2503155504993241551284026887086141.

a(n) is not of the form 4*k+5 (divisibility by 5) or 5*k+4 (divisibility by 11) or 9*k+4*(-1)^k (divisibility by 19).

Conjectures: a(n) is not of the form 7*k+4 or 8*k.

a(17) = 11071 mod 7 is 4, so the first half of the conjecture above is not true. - Robert Price, Sep 25 2019

a(23) = 78008 is divisible by 8, so the second half of the conjecture above is not true. - Robert Price, Sep 25 2019

a(24) > 2*10^5. - Robert Price, Apr 21 2020

Examples

			6 is in the sequence because 9^6 - 3^6 - 1 = 530711 is prime.
		

Crossrefs

Cf. numbers n such that k^(2*n)-k^n-1 is prime: A098845 (k=2), this sequence (k=3), A265482 (k=4), A265483 (k=5), A265484 (k=6), A265485 (k=7), A265486 (k=8), A265487 (k=9), A265383 (k=10), A265683 (k=16).

Programs

Magma

[n: n in [0..500] | IsPrime(9^n-3^n-1)];

Mathematica

Select[Range[1500], PrimeQ[9^# - 3^# - 1] &]

PARI

is(n)=ispseudoprime(9^n - 3^n - 1) \\ Anders Hellström, Dec 09 2015

Extensions

a(17)-a(23) from Robert Price, Sep 25 2019