A124621 -id:A124621 - cp's OEIS frontend

A217133 Numbers n such that 5^n + 8 is prime.

1, 95, 335, 3155, 28651, 91135

Offset: 1

Vincenzo Librandi, Oct 01 2012

Naturally these numbers are odd since (6-1)^(2n)+8 is divisible by 3. - Bruno Berselli, Oct 04 2012
a(7) > 10^5. - Robert Price, Feb 03 2014
a(7) > 5*10^5. - Tyler NeSmith, Apr 24 2022

Cf. A087885, A089142, A124621.

Select[Range[1, 15000, 2], PrimeQ[5^# + 8] &]

for(n=1, 5*10^3, if(isprime(5^n+8), print1(n", ")))

a(5)-a(6) from Robert Price, Feb 03 2014

A228028 Primes of the form 5^n + 4.

Original entry on oeis.org

5, 29, 15629, 9765629

Offset: 1

Vincenzo Librandi, Aug 11 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..8

Cf. A124621 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A228027 (k=4, h=9), A182330 (k=5, h=2), this sequence (k=5, h=4), A228029 (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A228030 (k=7, h=6), A228031 (k=7, h=10), A228032 (k=8, h=3), A228033 (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+4];

Select[Table[5^n + 4, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A243397 Numbers n such that 19^n+4 is prime.

Original entry on oeis.org

0, 1, 3, 21, 145, 273, 1425, 9613, 15711, 18445

Offset: 1

Felix Fröhlich, Jun 04 2014

No further terms up to 20000. - Felix Fröhlich, Oct 29 2014
No further terms up to 24000. - Felix Fröhlich, Jan 22 2015
No further terms up to 50000. - Michael S. Branicky, Oct 09 2024

Corresponding sequences for k^n+4: A058958 (k=3), A124621 (k=5), A096305 (k=7), A217384 (k=9), A137236 (k=13).

[n: n in [0..1000] | IsPrime(19^n+4)]; // Vincenzo Librandi, Oct 16 2014

Select[Range[0, 10000], PrimeQ[19^# + 4] &] (* Vincenzo Librandi, Oct 16 2014 *)

for(n=0, 10^5, if(ispseudoprime(19^n+4), print1(n, ", ")))

a(1)-a(2) prepended by N. J. A. Sloane, Jun 18 2014

a(9)-a(10) from Felix Fröhlich, Oct 16 2014

A247166 Numbers k such that 15^k+4 is prime.

Original entry on oeis.org

0, 1, 2, 7, 10, 39, 42, 201, 225, 551

Offset: 1

Felix Fröhlich, Dec 01 2014

No further terms up to 10000.

No further terms up to 10^5. - Tyler NeSmith, Jan 21 2021

Corresponding sequences for m^k+4: A058958 (m=3), A124621 (m=5), A096305 (m=7), A217384 (m=9), A137236 (m=13), A243397 (m=19).

[n: n in [0..300] | IsPrime(15^n+4)]; // Vincenzo Librandi, Dec 01 2015

a247166[n_Integer] := Select[Range[n], PrimeQ[15^# + 4] &]; a247166[10^4] (* Michael De Vlieger, Dec 03 2014 *)

for(n=0, 1e5, if(ispseudoprime(15^n+4), print1(n, ", ")))

Offset changed to 1 by Georg Fischer, Sep 26 2022

A217134 Numbers n such that 5^n - 8 is prime.

Original entry on oeis.org

2, 4, 10, 14, 88, 112, 140, 764, 3040, 11096, 24934, 25616, 54584, 93400

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

a(15) > 10^5. - Robert Price, Feb 03 2014

Cf. A059613, A087885, A089142, A109080, A124621, A165701.

Select[Range[2, 5000], PrimeQ[5^# - 8] &]

for(n=2, 5*10^3, if(isprime(5^n-8), print1(n", ")))

a(10)-a(14) from Robert Price, Feb 03 2014

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A253380 Numbers k such that 17^k + 4 is prime.

Original entry on oeis.org

0, 2, 6, 18, 7238

Offset: 1

Felix Fröhlich, Dec 31 2014

No further terms up to 10000.

No further terms up to 37200. - Michael S. Branicky, Mar 22 2023

			For k = 0: 17^0 + 4 = 5, which is prime, so 0 is a term of the sequence.
For k = 2: 17^2 + 4 = 293, which is prime, so 2 is a term of the sequence.

Corresponding sequences for k^n+4: A058958 (k=3), A124621 (k=5), A096305(k=7), A217384 (k=9), A137236 (k=13), A247166 (k=15), A243397 (k=19).

Select[Range@10^5, PrimeQ[17^# + 4] &] (* Michael De Vlieger, Jan 03 2015 *)

for(n=0, 1e5, if(ispseudoprime(17^n+4), print1(n, ", ")))

A378815 Numbers k such that 5^k + 64 is prime.

Original entry on oeis.org

2, 58, 170, 1402, 1774, 10802, 86342

Offset: 1

Robert Price, Dec 08 2024

			2 is a term because 5^2 + 64 = 89 is prime.

Cf. A089142, A124621, A217133.

Magma
```
[k: k in [0..1000] |IsPrime (5^k+64)];
```
Mathematica
```
Select[Range[0,5000],PrimeQ[5^#+64]&]
```

a(6) from Michael S. Branicky, Dec 17 2024

a(7) from Michael S. Branicky, Dec 23 2024

A378832 Numbers k such that 5^k + 68 is prime.

Original entry on oeis.org

1, 3, 7, 133, 331, 453, 10365

Offset: 1

Robert Price, Dec 08 2024

a(8) > 22000. - Matthew L. LaSelle, Feb 25 2025

a(8) > 100000. - Michael S. Branicky, Mar 28 2025

			3 is a term because 5^3 + 68 = 193 is prime.

Cf. A089142, A124621, A217133.

Magma
```
[k: k in [0..1000] |IsPrime (5^k+68)];
```
Mathematica
```
Select[Range[0,5000],PrimeQ[5^#+68]&]
```

a(7) from Michael S. Branicky, Dec 17 2024

A378866 Numbers k such that 5^k + 72 is prime.

Original entry on oeis.org

0, 2, 3, 118, 498, 1023, 4262, 6094, 6382, 26334, 56062

Offset: 1

Robert Price, Dec 09 2024

			3 is a term because 5^3 + 72 = 197 is prime.

Cf. A089142, A124621, A217133.

Magma
```
[k: k in [0..1000] |IsPrime (5^k+72)];
```
Mathematica
```
Select[Range[0,5000],PrimeQ[5^#+72]&]
```

a(8)-a(11) from Michael S. Branicky, Dec 19 2024

cp's OEIS Frontend

A217133 Numbers n such that 5^n + 8 is prime.

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A228028 Primes of the form 5^n + 4.

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A243397 Numbers n such that 19^n+4 is prime.

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Magma

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A247166 Numbers k such that 15^k+4 is prime.

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Magma

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A217134 Numbers n such that 5^n - 8 is prime.

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A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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PARI

A253380 Numbers k such that 17^k + 4 is prime.

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Mathematica

PARI

A378815 Numbers k such that 5^k + 64 is prime.

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Programs

Magma