A228032 -id:A228032 - cp's OEIS frontend

A228026 Primes of the form 4^k + 3.

7, 19, 67, 4099, 65539, 262147, 268435459, 1073741827, 19342813113834066795298819

Offset: 1

Vincenzo Librandi, Aug 11 2013

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

Vincenzo Librandi, Table of n, a(n) for n = 1..16

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).

[a: n in [0..200] | IsPrime(a) where a is  4^n+3];

Select[Table[4^n + 3, {n, 0, 200}], PrimeQ]

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

Cross-references corrected by Robert Price, Aug 01 2017

A228027 Primes of the form 4^k + 9.

Original entry on oeis.org

13, 73, 1033, 262153, 1073741833, 73786976294838206473, 4835703278458516698824713

Offset: 1

Vincenzo Librandi, Aug 11 2013

Subsequence of A104070. - Elmo R. Oliveira, Nov 28 2023

			262153 is a term because 4^9 + 9 = 262153 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..12

Cf. A000040, A217350 (corresponding k's).

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), A228026 (r=4, h=3), this sequence (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).

[a: n in [0..200] | IsPrime(a) where a is 4^n+9];

Select[Table[4^n + 9, {n, 0, 200}],PrimeQ]

a(n) = 4^A217350(n) + 9. - Elmo R. Oliveira, Nov 28 2023

Corrected cross-references - Robert Price, Aug 01 2017

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

A358079 Primes that can be written as 2^x + p where p is a prime and x is a multiple of p.

Original entry on oeis.org

11, 37, 67, 4099, 32771, 262147, 268435463, 1073741827, 36028797018963979, 18889465931478580854821, 151115727451828646838283, 19342813113834066795298819, 618970019642690137449562201, 316912650057057350374175801351, 85070591730234615865843651857942052871

Offset: 1

J. M. Bergot and Robert Israel, Oct 30 2022

nonn

			a(3) = 67 is a term because 67 = 2^6 + 3 where 67 and 3 are prime and 6 is divisible by 3.

Robert Israel, Table of n, a(n) for n = 1..32

Contains A057664 and A228032.

Cf. A358087.

R:= NULL: count:= 0:
for k from 1 while count < 15 do
  P:= sort(convert(numtheory:-factorset(k),list));
  for p in P do
    x:= 2^k+p;
    if isprime(x) then R:= R,x; count:= count+1; fi
od od:R;

cp's OEIS Frontend

A228026 Primes of the form 4^k + 3.

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Magma

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A228027 Primes of the form 4^k + 9.

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Magma

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A228028 Primes of the form 5^n + 4.

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Magma

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A358079 Primes that can be written as 2^x + p where p is a prime and x is a multiple of p.

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Maple