A239414 -id:A239414 - cp's OEIS frontend

A239415 Numbers n such that n^7-7 is prime.

60, 66, 132, 212, 242, 246, 290, 296, 312, 326, 380, 384, 446, 516, 524, 554, 654, 704, 740, 782, 834, 1026, 1086, 1142, 1154, 1172, 1182, 1214, 1424, 1430, 1464, 1482, 1494, 1500, 1524, 1604, 1682, 1686, 1752, 1794, 1796, 1844, 1854, 1940, 1952, 1980, 2000, 2010

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are even.

			60^7-7 = 2799359999993 is prime. Thus, 60 is a member of this sequence.

Cf. A028870, A153974, A239344, A239413, A239414.

Select[Range[2500],PrimeQ[#^7-7]&] (* Harvey P. Dale, Dec 21 2022 *)

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**7-7)}

A239416 Numbers n such that n^8-8 is prime.

Original entry on oeis.org

3, 7, 19, 39, 73, 75, 101, 107, 145, 147, 171, 213, 235, 247, 263, 285, 319, 353, 359, 369, 399, 443, 445, 521, 523, 557, 613, 675, 693, 707, 733, 781, 791, 805, 815, 829, 837, 879, 927, 943, 961, 999, 1033, 1097, 1103, 1109, 1129, 1137, 1141, 1155, 1157

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are odd.

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

Cf. A028870, A153974, A239413, A239414, A239415, A239417, A239418, A239345.

Select[Range[3,1200,2],PrimeQ[#^8-8]&] (* Harvey P. Dale, Jun 27 2014 *)

is(n)=isprime(n^8-8) \\ Charles R Greathouse IV, Feb 20 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**8-8)}

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

Original entry on oeis.org

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)}

A382246 Smallest number k such that k^n - 6 is prime.

Original entry on oeis.org

8, 3, 2, 5, 5, 5, 19, 85, 7, 5, 19, 275, 23, 43, 53, 455, 65, 23, 23, 175, 7, 65, 47, 295, 7, 143, 49, 115, 23, 355, 185, 305, 7, 55, 319, 85, 113, 25, 329, 505, 25, 187, 205, 25, 295, 437, 17, 2285, 7, 583, 35, 1375, 5, 7, 35, 895, 235, 277, 197, 695, 203, 145, 43, 35, 437, 215

Offset: 1

Jakub Buczak, Mar 19 2025

nonn

No term k in the sequence can be divisible by 2 or 3. Except for the special case a(1)-a(3), where the result of k^n - 6 is either the prime number 2 or 3.
If n is a multiple of 4, the only valid terms of k are those ending in a 5.
Empirical analysis suggests that the terms are typically prime or semiprime.

			a(1) = 8, because 8^1 - 6 = 2, which is prime.
a(4) = 5, because 5^4 - 6 = 619, which is prime.

Jakub Buczak, Table of n, a(n) for n = 1..500

Cf. A380905

Cf. A028879 (a(2)), A239414 (a(6)) for the first term.

a(n) = my(k=1); while (!isprime(k^n-6), k++); k; \\ Michel Marcus, Mar 19 2025

from sympy import isprime
def a(n):
    k = 1
    while (n>1 and k not in [2,3] and (k%2==0 or k%3==0)) or not isprime(k**n-6):
        k += 1
    return k

A239429 Numbers n such that n^6+6 and n^6-6 are prime.

Original entry on oeis.org

175, 12635, 18445, 30275, 32585, 38885, 41125, 46235, 53165, 71785, 74935, 92645, 108115, 117775, 121625, 146125, 151655, 173635, 184765, 191765, 196175, 204505, 208705, 229775, 237965, 241255, 243635, 246365, 283115, 335755, 344365, 345485, 352625, 353395, 354445

Offset: 1

Derek Orr, Mar 20 2014

nonn

All numbers are congruent to 35 mod 70.

Intersection of A109836 and A239414.

			175^6+6 = 28722900390631 is prime and 175^6-6 = 28722900390619 is prime. Thus, 175 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A109836, A239414.

Select[Range[35,360000,70],AllTrue[#^6+{6,-6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 22 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(6)

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.

A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

Original entry on oeis.org

3, 5, 23, 7, 433, 2447, 9377, 82597, 134687

Offset: 0

Jakub Buczak, Feb 07 2025

Terms must have an ending digit of 3, 5 or 7. If k ends in 1 or 9, then k^(2*3^n)-6 ends in a 5, which is not prime.

a(7) is the first composite term. - Michael S. Branicky, Feb 24 2025

			For n=0, k^(2*3^0) - 6 is prime for the first time at a(0) = k = 3.
For n=5, k^(2*3^5) - 6 is prime for the first time at a(5) = k = 2447.

Cf. Subsequence of A382246.
Cf. A008776, A025192.
Cf. A028879 (a(0)), A239414 (a(1)) for the first term.

a(n) = my(p=3,q=2*3^n); while (!ispseudoprime(p^q-6), p+=2); p; \\ Michel Marcus, Feb 08 2025

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(2) if k%10 in {3,5,7} and isprime(k**(2*3**n)-6))

a(7) from Michael S. Branicky, Feb 24 2025

a(8) from Georg Grasegger, Apr 17 2025

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.

cp's OEIS Frontend

A239415 Numbers n such that n^7-7 is prime.

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A239416 Numbers n such that n^8-8 is prime.

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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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A382246 Smallest number k such that k^n - 6 is prime.

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A239429 Numbers n such that n^6+6 and n^6-6 are 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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A380905 Smallest number k such that k^(2*3^n) - 6 is prime.

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