cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A253772 Numbers k such that 4^k + 13 is prime.

Original entry on oeis.org

1, 2, 4, 10, 19, 32, 40, 146, 566, 2054, 9967, 62639, 87814, 141092
Offset: 1

Views

Author

Vincenzo Librandi, Jan 12 2015

Keywords

Comments

Numbers of the form 4^n+k (for n>0) are never primes when k is even (obviously) or when k == -1 (mod 6): in the last case, in fact, (3+1)^n + 6*h-1 is divisible by 3. - Bruno Berselli, Oct 06 2015

Crossrefs

Cf. A104067.
Cf. Numbers k such that 4^k + d is prime: A089437 (d=3), A217349 (d=7), A217350 (d=9), this sequence (d=13), A253773 (d=15), A253774 (d=19), A262345 (d=21), A204388 (d=25), A262969 (d=27), A262971 (d=31), A262972 (d=33).

Programs

  • Magma
    [n: n in [0..2000] | IsPrime(4^n+13)];
    
  • Mathematica
    Select[Range[4000], PrimeQ[4^# + 13] &]
  • PARI
    is(n)=ispseudoprime(4^n+13) \\ Charles R Greathouse IV, Feb 17 2017

Formula

a(n) = A102634(n)/2. - Elmo R. Oliveira, Nov 12 2023

Extensions

a(11)-a(14) derived from A102634 by Robert Price, Sep 06 2015

A217354 Numbers n such that 8^n + 3 is prime.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 28, 76, 130, 370, 568, 713, 789, 790, 1334, 1354, 1849, 2913, 5729, 5740, 5978, 6908, 10618, 11918, 12748, 13449, 40850, 68654, 78442, 121040, 159948, 228526
Offset: 1

Views

Author

Vincenzo Librandi, Oct 02 2012

Keywords

Comments

3*A217354 is a subsequence of A057732. - Bruno Berselli, Oct 02 2012

Crossrefs

Programs

  • Mathematica
    Select[Range[5000], PrimeQ[8^# + 3] &]
  • PARI
    is(n)=ispseudoprime(8^n+3) \\ Charles R Greathouse IV, Feb 17 2017

Extensions

a(19)-a(32) are obtained from A057732; by Bruno Berselli, Oct 02 2012

A228026 Primes of the form 4^k + 3.

Original entry on oeis.org

7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819
Offset: 1

Views

Author

Vincenzo Librandi, Aug 11 2013

Keywords

Examples

			67 is a term because 4^3 + 3 = 67 is prime.
		

Crossrefs

Cf. A089437 (associated k).
Cf. Primes of the form r^k + h: A092506 (r=2, h=1), A057733 (r=2, h=3), A123250 (r=2, h=5), A104066 (r=2, h=7), A104070 (r=2, h=9), A057735 (r=3, h=2), A102903 (r=3, h=4), A102870 (r=3, h=8), A102907 (r=3, h=10), A290200 (r=4, h=1), this sequence (r=4, h=3), A228027 (r=4, h=9), A182330 (r=5, h=2), A228029 (r=5, h=6), A102910 (r=5, h=8), A182331 (r=6, h=1), A104118 (r=6, h=5), A104115 (r=6, h=7), A104065 (r=7, h=4), A228030 (r=7, h=6), A228031 (r=7, h=10), A228032 (r=8, h=3), A228033 (r=8, h=5), A144360 (r=8, h=7), A145440 (r=8, h=9), A228034 (r=9, h=2), A159352 (r=10, h=3), A159031 (r=10, h=7).

Programs

  • Magma
    [a: n in [0..200] | IsPrime(a) where a is  4^n+3];
  • Mathematica
    Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]

Formula

a(n) = 4^A089437(n) + 3. - Elmo R. Oliveira, Nov 14 2023

Extensions

Cross-references corrected by Robert Price, Aug 01 2017

A217349 Numbers k such that 4^k + 7 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 19, 22, 39, 44, 49, 63, 80, 87, 102, 107, 294, 305, 399, 463, 595, 599, 903, 944, 1324, 1727, 1755, 1932, 1935, 4485, 6165, 6665, 9438, 11169, 19859, 27503, 55392, 86235, 98217, 117855, 123640, 134204, 139660, 150437, 157634, 186475, 236129, 283248, 390142, 410178
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

The next terms are > 4.1*10^5. - Elmo R. Oliveira, Nov 29 2023

Examples

			For k = 14, 4^14 + 7 = 268435463 is prime.
		

Crossrefs

Cf. A057195, A059266, A089437, A104066 (associated primes).

Programs

  • Mathematica
    Select[Range[0, 5000], PrimeQ[4^# + 7] &]
  • PARI
    is(n)=ispseudoprime(4^n+7) \\ Charles R Greathouse IV, Jun 06 2017

Formula

a(n) = A057195(n)/2.

Extensions

Extended using A057195 terms by Michel Marcus, Aug 28 2015
a(51)-a(54) derived from A057195 by Elmo R. Oliveira, Nov 29 2023

A217350 Numbers k such that 4^k + 9 is prime.

Original entry on oeis.org

1, 3, 5, 9, 15, 33, 41, 335, 443, 671, 1197, 1355, 2247, 2639, 117293, 191099
Offset: 1

Views

Author

Vincenzo Librandi, Oct 01 2012

Keywords

Comments

The next terms are > 250000. - Robert Price, Oct 05 2015
Contains exactly the halved even terms of A057196.

Examples

			For k = 15, 4^15 + 9 = 1073741833 is prime.
		

Crossrefs

Cf. A057196, A089437 (similar sequence).

Programs

  • Magma
    [n: n in [0..700] | IsPrime(4^n+9)]; // Vincenzo Librandi, Oct 06 2015
    
  • Mathematica
    Select[Range[0, 5000], PrimeQ[4^# + 9] &]
  • PARI
    is(n)=ispseudoprime(4^n+9) \\ Charles R Greathouse IV, Jun 06 2017

Extensions

a(15)-a(16) derived from A057196 by Robert Price, Oct 05 2015

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

Views

Author

Eric Chen, Jun 04 2018

Keywords

Comments

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.

Crossrefs

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)
--------------------------------------------------------------------
4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 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)).

Programs

  • PARI
    a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))
Showing 1-6 of 6 results.