A125260 -id:A125260 - cp's OEIS frontend

A067200 Numbers m such that m^3 + 2 is prime.

0, 1, 3, 5, 29, 45, 63, 65, 69, 71, 83, 105, 113, 123, 129, 143, 153, 171, 173, 189, 209, 215, 219, 231, 243, 245, 249, 263, 291, 299, 305, 311, 341, 363, 369, 395, 419, 425, 431, 435, 473, 483, 491, 495, 501, 503, 509, 515, 533, 549, 555, 561, 575, 579, 639

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A144953.

Other sequences of the type "Numbers m such that m^k + k - 1 is prime": A000040 (k=1), A005574 (k=2), this sequence (k=3), A125259 (k=4), A125260 (k=5), A125261 (k=6), A125262 (k=7), A125263 (k=8), A125264 (k=10), A125265 (k=11).

lst={}; Do[If[PrimeQ[n^3+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[0,700],PrimeQ[#^3+2]&] (* Harvey P. Dale, Jul 27 2025 *)

is(n)=isprime(n^3 + 2) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = (A144953(n) - 2)^(1/3). - Zak Seidov, Sep 16 2013

A125259 Numbers k such that k^4 + 3 is prime.

Original entry on oeis.org

0, 2, 8, 16, 22, 26, 28, 34, 44, 62, 68, 76, 82, 92, 104, 110, 118, 128, 134, 166, 184, 202, 212, 266, 286, 296, 314, 328, 350, 356, 376, 406, 428, 436, 460, 470, 506, 520, 532, 562, 580, 638, 650, 652, 680, 692, 722, 734, 740, 778, 812, 820, 824, 862, 896, 908

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..5860

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), this sequence (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)...

Select[Range[0,1000],PrimeQ[#^4+3]&] (* Harvey P. Dale, Aug 02 2023 *)

isok(n, k=4) = isprime(n^k + k - 1); \\ Michel Marcus, Oct 11 2013

A125265 Numbers k such that k^11 + 10 is prime.

Original entry on oeis.org

1, 7, 19, 21, 33, 69, 153, 157, 193, 253, 379, 391, 439, 543, 549, 559, 579, 609, 879, 937, 939, 993, 1063, 1083, 1107, 1119, 1191, 1209, 1267, 1281, 1287, 1333, 1537, 1617, 1797, 1819, 1923, 1971, 1987, 1989, 2041, 2061, 2073, 2101, 2103, 2343, 2373

Offset: 1

Zak Seidov, Nov 26 2006

nonn

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

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10).

lst={};k=11;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[2500],PrimeQ[#^11+10]&] (* Harvey P. Dale, Jul 01 2015 *)

is(n)=isprime(n^11+10) \\ Charles R Greathouse IV, Feb 17 2017

A125261 Numbers k such that k^6 + 5 is prime.

Original entry on oeis.org

0, 18, 24, 114, 204, 216, 222, 246, 276, 312, 318, 372, 384, 426, 438, 468, 498, 582, 618, 654, 822, 888, 948, 984, 1182, 1188, 1272, 1278, 1284, 1374, 1446, 1536, 1674, 1782, 1788, 1794, 1806, 1812, 1896, 2034, 2058, 2088, 2124, 2154, 2232, 2238, 2376

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..2254

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), this sequence(j=6), A125262 (j=7), A125263 (j=8), A125264 (j=10), A125265 (j=11)..

lst={};k=6;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,2500],PrimeQ[#^6+5]&] (* Harvey P. Dale, Aug 29 2012 *)

is(n)=isprime(n^6+5) \\ Charles R Greathouse IV, Feb 17 2017

A125263 Numbers k such that k^8 + 7 is prime.

Original entry on oeis.org

0, 2, 4, 10, 66, 68, 86, 88, 134, 146, 200, 216, 250, 276, 306, 310, 410, 422, 472, 492, 506, 516, 538, 548, 550, 594, 638, 716, 746, 758, 862, 888, 942, 954, 964, 982, 992, 998, 1000, 1016, 1020, 1034, 1108, 1164, 1192, 1234, 1338, 1342, 1350, 1374, 1390

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..4683

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), this sequence (j=8), A125264 (j=10), A125265 (j=11)...

lst={};k=8;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[0,1500],PrimeQ[#^8+7]&] (* Harvey P. Dale, Sep 04 2024 *)

is(n)=isprime(n^8+7) \\ Charles R Greathouse IV, Feb 17 2017

A125264 Numbers k such that k^10 + 9 is prime.

Original entry on oeis.org

2, 8, 238, 310, 338, 442, 542, 688, 698, 872, 920, 1198, 1330, 1382, 1538, 1558, 1678, 1702, 1712, 1768, 1882, 2032, 2080, 2102, 2260, 2312, 2408, 2440, 2540, 2642, 3112, 3170, 3188, 3268, 3320, 3580, 3740, 3742, 3770, 3980, 4028, 4048, 4148, 4292, 4472

Offset: 1

Zak Seidov, Nov 26 2006

nonn

Carmine Suriano, Table of n, a(n) for n = 1..1524

Other sequences of the type "Numbers k such that k^j + j - 1 is prime": A000040 (j=1), A005574 (j=2), A067200 (j=3), A125259 (j=4), A125260 (j=5), A125261 (j=6), A125262 (j=7), A125263 (j=8), this sequence (j=10), A125265 (j=11).

lst={};k=10;Do[If[PrimeQ[n^k+k-1], AppendTo[lst, n]], {n, 0, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 10 2008 *)
Select[Range[5000],PrimeQ[#^10+9]&] (* Harvey P. Dale, Nov 17 2014 *)

is(n)=isprime(n^10+9) \\ Charles R Greathouse IV, Feb 17 2017

A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

Original entry on oeis.org

7, 5503, 21013, 301123, 303613, 420037, 469363, 679153, 771427, 991957, 999667, 1524763, 1707367, 2030653, 2333083, 2540563, 2552713, 2710933, 3009967, 3378103, 3441817, 3592213, 4419937, 4704613, 4840723, 5177797, 5691547, 6227587, 6275887, 6395677, 6595597, 6597163

Offset: 1

Zak Seidov, Aug 28 2014

nonn

For even k > 2, 4 + n^k is prime only for n = 1.
From Derek Orr, Aug 28 2014 (edited by Danny Rorabaugh, Apr 19 2015): (Start)
4+p^4 is composite for all primes p. For p = 2, 4+p^4 = 20 is composite. To prove it for odd primes, consider S(n) = 4+(2*n+1)^4. S(n) == 0 (mod 5) unless n == 2 (mod 5). If n == 2 (mod 5), then 2*n+1 == 0 (mod 5), which is only prime for n = 2; this gives p = 5 and 4+5^4 = 629 is composite. For other odd primes p, 4+p^4 is greater than 5 and divisible by 5.
4+p^(4*m) is also composite for any prime p and integer m > 0. For each m, the proof is the same as above.
(End)
All terms are == {3,7} (mod 10). - Zak Seidov, Aug 29 2014

			From _K. D. Bajpai_, Jan 20 2015: (Start)
a(2) = 5503:
4 + 5503 = 5507;
4 + 5503^2 = 30283013;
4 + 5503^3 = 166647398531;
4 + 5503^5 = 5046584669419727747;
all five are prime.
(End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007591, A073573, A125260, A172367.
Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime is A253937. - K. D. Bajpai, Jan 20 2015
The subsequence with 4+p^7 also prime is A246562. - Danny Rorabaugh, Apr 19 2015

[p: p in PrimesUpTo(2*10^7) | IsPrime(4+p) and IsPrime(4+p^2) and IsPrime(4+p^3) and IsPrime(4+p^5)]; // Vincenzo Librandi, Apr 19 2015

k=4; Select[Prime[Range[1,500000]], PrimeQ[k+#]&&PrimeQ[k+#^2] &&PrimeQ[k+#^3] &&PrimeQ[k+#^5]&]  (*K. D. Bajpai, Jan 20 2015 *)

for(n=1, 6000000, if(isprime(n) && isprime(4+n) && isprime(4+n^2) && isprime(4+n^3) && isprime(4+n^5), print1(n, ", "))) \\ Colin Barker, Aug 28 2014

p=7; forprime(q=11, 1e8, if(q-p==4 && isprime(4+p^2) && isprime(4+p^3) && isprime(4+p^5), print1(p, ", ")); p=q) \\ Charles R Greathouse IV, Aug 28 2014

from sympy import prime, isprime
A246519_list = [p for p in (prime(n) for n in range(1,10**5)) if all([isprime(4+p**z) for z in (1,2,3,5)])]
# Chai Wah Wu, Sep 08 2014

A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

Original entry on oeis.org

7, 469363, 2552713, 3378103, 6595597, 6629683, 39837517, 46024063, 46167307, 97371007, 97629403, 105528217, 136983307, 169483033, 202953613, 213792193, 216520987, 216738043, 221705647, 304033927, 317502193, 359133553

Offset: 1

Zak Seidov, Aug 29 2014

nonn

All terms are == {3, 7} mod 10. Subsequence of A246519.

Zak Seidov, Table of n, a(n) for n = 1..60

Cf. A007591, A073573, A125260, A172367, A246519.

Select[Prime[Range[193*10^5]],AllTrue[#^{1,2,3,5,7}+4,PrimeQ]&] (* Harvey P. Dale, Sep 07 2024 *)

forprime(p=1,10^9,if(ispseudoprime(4+p) && ispseudoprime(4+p^2) && ispseudoprime(4+p^3) && ispseudoprime(4+p^5) && ispseudoprime(4+p^7), print1(p,", "))) \\ Derek Orr, Aug 30 2014

A253937 Primes p such that 4+p^7, 4+p^9 and 4+p^11 are also prime.

Original entry on oeis.org

82609, 1032607, 1859479, 2158447, 4952173, 5009593, 5828353, 6779833, 11316859, 11370727, 12786157, 13872853, 14117053, 15082783, 15645697, 15935989, 16715623, 20102569, 21310603, 22106569, 22164253, 23674597, 26012953, 26325613, 29592919, 30086347, 30306637

Offset: 1

K. D. Bajpai, Jan 19 2015

nonn

			a(1) = 82609:
4 + 82609^7 = 26253762656881427836948640304009173;
4 + 82609^9 = 179162157925737357103123335151825463343651893;
4 + 82609^11 = 1222646797417942588836172615268162579679296234658008213;
all four are prime.

K. D. Bajpai, Table of n, a(n) for n = 1..720

Cf. A000040, A125260, A246519, A246562.

Select[Prime[Range[1, 2000000]], PrimeQ[4 + #^7] && PrimeQ[4 + #^9] && PrimeQ[4 + #^11] &]

forprime(p=1, 1e7, if(isprime(4+p^7) && isprime(4+p^9) && isprime(4+p^11), print1(p,", ")))

A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

Original entry on oeis.org

3, 3, 3, 1, 7, 3, 7, 1, 3, 3, 9, 1, 33, 7, 9, 1, 43, 17, 27, 1, 9, 3, 7, 1, 55, 47, 285, 1, 27, 3, 39, 1, 43, 117, 163, 1, 63, 255, 15, 1, 87, 3, 43, 1, 187, 77, 37, 1, 105, 45, 25, 1, 99, 305, 79, 1, 3, 27, 903, 1, 127, 293, 255, 1, 27, 27, 435, 1, 207, 143, 127, 1, 117, 295, 1159, 1, 477

Offset: 1

Zak Seidov, Aug 29 2014

nonn

If n is a multiple of 4 then 4 + m^n is prime iff m = 1.

4 + m^(4*x) = (m^(2*x)-2*m^x+2) * (m^(2*x)+2*m^x+2). - Jens Kruse Andersen, Sep 02 2014

Zak Seidov and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 200 terms from Zak Seidov)

Cf. A007591, A073573, A125260, A172367, A246519.

a(n)=if(n%4==0,return(1));m=2;while(!ispseudoprime(4+m^n),m++);return(m)
vector(100,n,a(n)) \\ Derek Orr, Aug 29 2014

cp's OEIS Frontend

A067200 Numbers m such that m^3 + 2 is prime.

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A125259 Numbers k such that k^4 + 3 is prime.

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A125265 Numbers k such that k^11 + 10 is prime.

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A125261 Numbers k such that k^6 + 5 is prime.

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A125263 Numbers k such that k^8 + 7 is prime.

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A125264 Numbers k such that k^10 + 9 is prime.

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A246519 Primes p such that 4+p, 4+p^2, 4+p^3 and 4+p^5 are all prime.

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A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

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