A153974 -id:A153974 - cp's OEIS frontend

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

3, 9, 15, 21, 33, 117, 237, 273, 303, 309, 387, 429, 441, 447, 513, 561, 573, 609, 807, 897, 1035, 1071, 1113, 1143, 1233, 1239, 1311, 1563, 1611, 1617, 1737, 1749, 1827, 1839, 1953, 2133, 2211, 2283, 2589, 2715, 2721, 2955, 3081, 3093, 3453, 3549, 3555, 3621, 3723, 3807

Offset: 1

John L. Drost, Jun 19 2005

Since x^2 + 2 is divisible by 3 unless x is divisible by 3, all elements are 3 mod 6.

Intersection of A067201 and A028870. - Robert Israel, Sep 11 2014

			21 is on the list since 21^2 - 2 = 439 and 21^2 + 2 = 443 are primes.

David Wells, Prime Numbers, John Wiley and Sons, 2005, p. 219 (article:'Siamese primes')

Nathaniel Johnston, Table of n, a(n) for n = 1..8750
Raymond A. Beauregard and E. R. Suryanarayan, Square-plus-two primes, Mathematical Gazette 85(502) 90-1.

Subsequence of A016945.

Cf. A016945, A028870, A028873, A038599, A067201, A153974.

[n: n in [3..3600 by 6] | IsPrime(n^2-2) and IsPrime(n^2+2)];  // Bruno Berselli, Apr 15 2011

select(n -> isprime(n^2-2) and isprime(n^2+2), [seq(6*i+3,i=0..1000)]); # Robert Israel, Sep 11 2014

Select[Range[5000], PrimeQ[#^2 - 2] && PrimeQ[#^2 + 2] &] (* Alonso del Arte, Sep 11 2014 *)

is(n)=isprime(n^2-2)&&isprime(n^2+2) \\ Charles R Greathouse IV, Jul 02 2013

Terms corrected by Charles R Greathouse IV, Sep 11 2014

A239414 Numbers k such that k^6 - 6 is prime.

Original entry on oeis.org

5, 7, 17, 37, 113, 137, 157, 173, 175, 203, 223, 227, 295, 337, 395, 407, 475, 487, 503, 535, 605, 617, 707, 743, 797, 833, 857, 863, 865, 877, 905, 943, 947, 965, 973, 995, 1037, 1043, 1057, 1103, 1217, 1243, 1247, 1277, 1295, 1337, 1357, 1363, 1375, 1403

Offset: 1

Derek Orr, Mar 17 2014

nonn

Note that all the numbers in this sequence are odd.

			5^6 - 6 = 15619 is prime. Thus, 5 is a member of this sequence.

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

Cf. A028870, A109836, A153974.

Select[Range[3,1501,2],PrimeQ[#^6-6]&] (* Harvey P. Dale, Jul 24 2016 *)

is(n)=isprime(n^6-6) \\ Charles R Greathouse IV, Feb 17 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**6-6)}

A239413 Numbers n such that n^5-5 is prime.

Original entry on oeis.org

4, 12, 16, 42, 102, 124, 132, 144, 184, 232, 274, 288, 306, 316, 336, 352, 406, 438, 478, 582, 606, 622, 706, 742, 754, 762, 814, 832, 916, 922, 964, 984, 996, 1026, 1044, 1072, 1086, 1096, 1156, 1174, 1204, 1258, 1272, 1366, 1408, 1416, 1428, 1432, 1456

Offset: 1

Derek Orr, Mar 17 2014

Note that all the numbers in this sequence are even.

There is no sequence "Numbers n such that n^4-4 is prime." since n^4 - 4 = (n^2 + 2)(n^2 - 2). - Michael B. Porter, Mar 18 2014

			4^5-5 = 1019 is prime. Thus, 4 is a member of this sequence.

Cf. A028870, A153974, A239343.

Select[Range[1500],PrimeQ[#^5-5]&] (* Harvey P. Dale, Dec 30 2019 *)

is(n)=isprime(n^5-5) \\ Charles R Greathouse IV, Feb 17 2017

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**5-5)}

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

Original entry on oeis.org

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)}

A153975 Values of n such that n^2-3 and n^2+3 are both prime.

Original entry on oeis.org

4, 8, 10, 14, 64, 92, 112, 140, 146, 172, 218, 298, 304, 322, 326, 340, 350, 356, 416, 440, 470, 508, 554, 560, 580, 626, 634, 652, 668, 686, 694, 704, 728, 736, 746, 770, 806, 818, 868, 892, 920, 1054, 1082, 1102, 1130, 1156, 1196, 1256, 1264, 1378, 1418

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Intersection of A028873 and A049422. - Zak Seidov, Oct 12 2014

			4^2 - 3 = 13 and 4^2 + 3 = 19 are both primes, so 4 is in the sequence.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A028870, A028873, A038599, A049422, A108701, A153974.

[n: n in [1..1400] | IsPrime(n^2-3) and IsPrime(n^2+3)]; // Vincenzo Librandi, Oct 12 2014

Select[Range[1500], PrimeQ[#^2 - 3] && PrimeQ[#^2 + 3] &] (* Vincenzo Librandi, Oct 12 2014 *)

is(n) = isprime(n^2-3) && isprime(n^2+3); \\ Altug Alkan, Sep 01 2016

Incorrect term 0 removed and Mma edited by Zak Seidov, Oct 12 2014

A241808 Numbers k such that (2*k)^3 - 3 is prime.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 13, 17, 19, 20, 37, 40, 53, 55, 58, 62, 68, 79, 89, 92, 95, 103, 112, 115, 119, 128, 137, 140, 158, 160, 169, 170, 193, 205, 214, 223, 229, 232, 235, 242, 248, 265, 272, 275, 278, 295, 313, 317, 322, 323, 337, 355, 359, 364

Offset: 1

Gerasimov Sergey, Apr 29 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006254, A153974.

[n: n in [1..500] | IsPrime((2*n)^3 - 3)]; // Juri-Stepan Gerasimov, May 12 2014

Select[Range[400], PrimeQ[(2*#)^3 - 3] &] (* Amiram Eldar, Aug 29 2020 *)

is(n)=isprime((2*n)^3-3) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A153974(n)/2. - R. J. Mathar, May 14 2014

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.

cp's OEIS Frontend

A108701 Values of n such that n^2-2 and n^2+2 are both prime.

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A239414 Numbers k such that k^6 - 6 is prime.

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A239413 Numbers n such that n^5-5 is prime.

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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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