A028876 -id:A028876 - cp's OEIS frontend

A352804 a(n) = A028876(n)/2; numbers k such that 4*k^2 - 5 is prime.

2, 3, 4, 6, 7, 8, 11, 12, 16, 17, 18, 19, 21, 22, 23, 26, 29, 32, 36, 37, 39, 41, 47, 51, 56, 58, 61, 66, 72, 76, 82, 83, 84, 87, 88, 91, 92, 93, 94, 99, 102, 106, 111, 113, 116, 117, 118, 126, 131, 132, 139, 142, 144, 146, 149, 151, 159, 162, 171, 177, 179

Offset: 1

Jianing Song, Apr 04 2022

A028876 with common factor 2 removed.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A028876, A028877 (the resulting primes), A352805.

Mathematica
```
Select[Range[200], PrimeQ[4*#^2 - 5] &]
```
PARI
```
isA352804(n)=isprime(4*n^2-5)
```

A028877 Primes of form k^2 - 5.

Original entry on oeis.org

11, 31, 59, 139, 191, 251, 479, 571, 1019, 1151, 1291, 1439, 1759, 1931, 2111, 2699, 3359, 4091, 5179, 5471, 6079, 6719, 8831, 10399, 12539, 13451, 14879, 17419, 20731, 23099, 26891, 27551, 28219, 30271, 30971, 33119, 33851, 34591, 35339, 39199, 41611, 44939, 49279

Offset: 1

Patrick De Geest

These numbers are prime in Z but not in Z[sqrt(5)] nor in Z[phi] (where phi is the golden ratio), since (k - sqrt(5))(k + sqrt(5)) = ((k + 1) - 2*phi)((k - 1) + 2*phi) = k^2 - 5. - Alonso del Arte, Aug 27 2013

			31 is in the sequence as it is equal to 6^2 - 5.
59 is in the sequence since it is equal to 8^2 - 5.
95 is not in the sequence though it does equal 10^2 - 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Patrick De Geest, Palindromic Quasipronics of the form n(n+x).
Eric Weisstein's World of Mathematics, Near-Square Prime.

Cf. A028875 (superset), A028876.

[a: n in [1..300] | IsPrime(a) where a is n^2-5]; // Vincenzo Librandi, Dec 01 2011

Select[Table[n^2 - 5, {n, 200}], PrimeQ] (* Harvey P. Dale, Jan 17 2011 *)

a(n) = A028875(A028876(n)). - Elmo R. Oliveira, Feb 22 2025

A296507 Numbers m such that m^2 - 13 is a prime.

Original entry on oeis.org

4, 6, 12, 18, 24, 30, 36, 54, 72, 84, 90, 96, 102, 114, 120, 138, 168, 186, 198, 204, 210, 216, 228, 240, 276, 294, 318, 330, 354, 360, 372, 378, 402, 414, 438, 444, 456, 480, 498, 504, 588, 600, 612, 618, 630, 636, 666, 678, 690, 714, 720, 726, 732, 738, 762

Offset: 1

Zak Seidov, Dec 13 2017

nonn

All terms except 4 are divisible by 6. - Robert Israel, Dec 13 2017

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A028870, A028873, A028876, A028882, A090696.

select(n -> isprime(n^2-13), 2*[$2..10^4]); # Robert Israel, Dec 13 2017

Reap[m=4;Do[If[PrimeQ[m^2-13],Sow[m]];m=m+2,{1000}]][[2,1]]
Select[Range[800],PrimeQ[#^2-13]&] (* Harvey P. Dale, Mar 06 2023 *)

isok(n) = isprime(n^2-13); \\ Michel Marcus, Dec 14 2017

A309726 Numbers k such that k^2 - 12 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 25, 29, 35, 41, 49, 53, 59, 61, 79, 85, 91, 95, 97, 103, 107, 113, 119, 121, 137, 139, 145, 149, 163, 169, 173, 179, 181, 185, 191, 205, 209, 227, 233, 235, 245, 251

Offset: 1

Daniel Starodubtsev, Aug 14 2019

nonn

All terms are odd and not divisible by 3.

			11 is in the sequence because 11^2 - 12 = 109, which is prime.

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

Cf. A028870, A028873, A028876, A028879, A028882, A028885, A186815, A090696, A296507.

Cf. A056927.

Select[Range[5,301,2],PrimeQ[#^2-12]&] (* Harvey P. Dale, Dec 23 2019 *)

select(n->isprime(n^2-12), [1..1000]) \\ Andrew Howroyd, Aug 14 2019

If A056927(k) = 12, then k is a term. - A.H.M. Smeets, Aug 15 2019

cp's OEIS Frontend

A352804 a(n) = A028876(n)/2; numbers k such that 4*k^2 - 5 is prime.

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A028877 Primes of form k^2 - 5.

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Magma

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A296507 Numbers m such that m^2 - 13 is a prime.

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Maple

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A309726 Numbers k such that k^2 - 12 is prime.

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