A239416 -id:A239416 - cp's OEIS frontend

A239417 Numbers n such that n^9-9 is prime.

2, 62, 86, 88, 116, 152, 266, 292, 310, 326, 338, 356, 406, 436, 466, 470, 518, 550, 568, 616, 626, 650, 688, 700, 722, 812, 850, 926, 956, 992, 1058, 1076, 1126, 1186, 1252, 1430, 1550, 1570, 1642, 1672, 1682, 1766, 1808, 1852, 1868, 1888, 2138, 2210, 2306

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all numbers in this sequence are even.

			2^9-9 = 503 is prime. Thus, 2 is a member of this sequence.

Cf. A028870, A153974, A239413, A239414, A239415, A239416, A239418, A239346.

is(n)=isprime(n^9-9) \\ Charles R Greathouse IV, Feb 20 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**9-9)}

A239418 Numbers n such that n^10 - 10 is prime.

Original entry on oeis.org

21, 201, 267, 321, 369, 459, 537, 651, 669, 699, 723, 753, 1071, 1113, 1197, 1203, 1209, 1323, 1401, 1503, 1587, 1647, 1773, 1791, 1797, 1917, 1941, 2007, 2139, 2223, 2427, 2493, 2613, 2733, 2769, 2787, 2847, 3147, 3249, 3267, 3297, 3399, 3423, 3441, 3771

Offset: 1

Derek Orr, Mar 17 2014

nonn

All of the numbers in this sequence are odd multiples of 3 and, thus, congruent to 3 (mod 6).

The tenth powers modulo 6 are 1, 4, 3, 4, 1, 0, ... (A070431). Subtracting 10 (still modulo 6), we get 3, 0, 5, 0, 3, 2, ... which means that only n = 3 mod 6 can produce a potential prime p = 5 mod 6.

			21^10 - 10 = 16679880978191 is prime. Thus, 21 is a member of this sequence.

Cf. A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239347.

Select[Range[1000], PrimeQ[#^10 - 10] &] (* Alonso del Arte, Mar 18 2014 *)

is(n)=isprime(n^10-10) \\ Charles R Greathouse IV, Feb 20 2017

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

A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

Original entry on oeis.org

3, 2, 2, 0, 4, 5, 60, 3, 2, 21, 28, 5, 2, 199, 28, 0, 234, 11, 2, 3, 2, 159, 10, 31, 68, 145, 0, 69, 186, 163, 32, 253, 26, 261, 4, 0, 8, 11, 62, 3, 22, 43, 6, 7, 8, 945, 76, 7, 116, 129, 382, 93, 330, 361, 2, 555, 224, 1359, 78, 29, 62, 39, 110, 0, 1032, 37, 462, 29

Offset: 1

Derek Orr, Mar 20 2014

nonn

If n is of the form (pk)^p for some k and some prime p, then a(n) = 0 (See A097764).

			1^1-1 = 0 is not prime. 2^1-1 = 1 is not prime. 3^1-1 = 2 is prime. Thus, a(1) = 3.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418.

import sympy
from sympy import isprime
def TwoMin(x):
  for k in range(1,5000):
    if isprime(k**x-x):
      return k
x = 1
while x < 100:
  print(TwoMin(x))
  x += 1

a(A097764(n)) = 0 for all n.

A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

Original entry on oeis.org

4, 3, 2, 0, 42, 175, 66, 3, 2, 4983, 1770, 55055, 28686, 18765, 8456, 0, 594, 128345, 136080, 81, 92, 1163409, 18810, 10415, 11754, 3855, 0, 86043, 38880, 17639, 26088, 37293, 5540, 612015, 6876, 0, 44220, 130425, 110, 9292527, 1004850, 1812149, 442404, 1007445, 570658

Offset: 1

Derek Orr, Mar 20 2014

nonn

a(n) = 0 iff n is of the form (pk)^p for some k and some prime p (See A097764).

gcd(n,a(n)) = 1 for all a(n) > 0.

			1^1 +/- 1 = 2 and 0 are not both primes. 2^1 +/- 1 = 3 and 1 are not both primes. 3^1 +/- 1 = 4 and 2 are not both primes. 4^1 +/- 1 = 5 and 3 are both primes. Thus a(1) = 4.

Cf. A072883, A028870, A153974, A239413, A239414, A239415, A239416, A239417, A239418, A239474.

a(n)=for(k=1,10^7,if(ispseudoprime(k^n-n)&&ispseudoprime(k^n+n),return(k)))
n=1;while(n<100,print1(a(n),", ");n++)

import sympy
from sympy import isprime
def TwoBoth(x):
  for k in range(1,10**7):
    if isprime(k**x+x) and isprime(k**x-x):
      return k
x = 1
while x < 100:
  if TwoBoth(x) != None:
    print(TwoBoth(x))
  else:
    print(0)
  x += 1

a(A097764(n)) = 0 for all n.

A239503 Numbers n such that n^8+8 and n^8-8 are prime.

Original entry on oeis.org

3, 1515, 1689, 3327, 4461, 4641, 4965, 5043, 5583, 5709, 6183, 7089, 9291, 9369, 9699, 10125, 11109, 14175, 15081, 18393, 20295, 26955, 27009, 27219, 29067, 30513, 30807, 35355, 35889, 36003, 37935, 40107, 43461, 48045, 49005, 51783, 53289, 55527, 58833, 61203

Offset: 1

Derek Orr, Mar 20 2014

nonn

All numbers are congruent to 3 mod 6.

Intersection of A239345 and A239416.

			3^8+8 = 6569 is prime and 3^8-8 = 6553 is prime. Thus, 3 is a member of this sequence.

Cf. A239345, A239416.

Select[Range[3,62000,6],AllTrue[#^8+{8,-8},PrimeQ]&](* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 07 2020 *)

import sympy
from sympy import isprime
def TwoBoth(x):
  for k in range(10**6):
    if isprime(k**x+x) and isprime(k**x-x):
      print(k)
TwoBoth(8)

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A239417 Numbers n such that n^9-9 is prime.

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A239418 Numbers n such that n^10 - 10 is prime.

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A239474 Smallest k >= 1 such that k^n-n is prime. a(n) = 0 if no such k exists.

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A239475 Least number k such that k^n + n and k^n - n are both prime, or 0 if no such number exists.

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A239503 Numbers n such that n^8+8 and n^8-8 are prime.

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Mathematica

Python