A162295 -id:A162295 - cp's OEIS frontend

A162294 Numbers k such that k^3-k^2-k-1 is prime.

4, 6, 8, 12, 16, 22, 28, 34, 44, 50, 54, 56, 58, 76, 78, 88, 110, 112, 118, 134, 138, 156, 162, 166, 168, 170, 188, 190, 200, 204, 208, 210, 226, 230, 236, 244, 250, 268, 274, 302, 310, 314, 322, 324, 340, 344, 356, 364, 368, 378, 382, 390, 398, 400, 420, 424

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 30 2009

nonn

			k=4 is in the sequence because 4^3-4^2-4-1=43 is prime.
k=6 is in the sequence because 6^3-6^2-6-1=173 is prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A087908, A111501, A162291, A162293, A162295 (corresponding primes).

lst={};Do[p=n^3-n^2-n-1;If[PrimeQ[p],AppendTo[lst,n]],{n,2,6!}];lst

k^3-k^2-k-1 = A162295(n), where k=a(n).

Sum_{i=1..n} a(i) = Sum_{i=1..n} i * ( pi(i^3 - i^2 - i - 1) - pi(i^3 - i^2 - i - 2) ). - Wesley Ivan Hurt, May 24 2013

Edited by R. J. Mathar, Jul 02 2009

A237528 Numbers n of the form p^3-p^2-p-1 (for prime p) such that n^3-n^2-n-1 is prime.

Original entry on oeis.org

23518, 146014, 486718, 564814, 3285598, 4629406, 7151614, 11645326, 22584814, 29983198, 31754206, 64319998, 355897438, 745319086, 864614014, 1304555614, 2334990526, 2903803726, 3447511198, 3934332718, 4194050014, 4596374014, 5838479998, 6866219998

Offset: 1

Derek Orr, Feb 09 2014

nonn

All numbers are congruent to 4 mod 10, 6 mod 10, or 8 mod 10.

			23518 = 29^3-29^2-29-1 (29 is prime) and 23518^3-23518^2-23518-1 = 13007166227989 is prime. Thus, 23518 is a member of this sequence.

Cf. A162295.

f[n_] := n^3 - n^2 - n - 1; f[ Select[ Prime@ Range[2,740],PrimeQ@ f@ f@#&]] (* Robert G. Wilson v, Mar 07 2014 *)

s=[]; forprime(p=2, 40000, n=p^3-p^2-p-1; if(isprime(n^3-n^2-n-1), s=concat(s, n))); s \\ Colin Barker, Feb 10 2014

import sympy
from sympy import isprime
{print(n**3-n**2-n-1) for n in range(10**4) if isprime(n) and isprime((n**3-n**2-n-1)**3-(n**3-n**2-n-1)**2-(n**3-n**2-n-1)-1)}

A230027 Primes p such that f(f(p)) is prime, where f(x) = x^3-x^2-x-1.

Original entry on oeis.org

29, 53, 79, 83, 149, 167, 193, 227, 283, 311, 317, 401, 709, 907, 953, 1093, 1327, 1427, 1511, 1579, 1613, 1663, 1801, 1901, 1987, 2027, 2029, 2293, 2341, 2741, 2887, 3083, 3163, 3229, 3329, 3511, 3733, 4007, 4127, 4153, 4337, 4789, 5531

Offset: 1

Derek Orr, Feb 23 2014

nonn

			29 is prime and (29^3-29^2-29-1)^3 - (29^3-29^2-29-1)^2 - (29^3-29^2-29-1) - 1 = 13007166227989 is prime. Thus, 29 is a member of this sequence.

Cf. A000040, A237528, A162295.

f[n_] := n^3 - n^2 - n - 1; Select[ Prime@ Range[2, 740], PrimeQ@ f@ f@# &] (* Robert G. Wilson v, Mar 07 2014 *)

from sympy import isprime
def f(x):
  return x**3-x**2-x-1
{print(p) for p in range(10**4) if isprime(p) and isprime(f(f(p)))}

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A162294 Numbers k such that k^3-k^2-k-1 is prime.

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A237528 Numbers n of the form p^3-p^2-p-1 (for prime p) such that n^3-n^2-n-1 is prime.

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A230027 Primes p such that f(f(p)) is prime, where f(x) = x^3-x^2-x-1.

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