A110589 -id:A110589 - cp's OEIS frontend

A118940 Primes p such that (p^2+7)/8 is prime.

3, 7, 17, 23, 41, 47, 71, 89, 103, 113, 127, 137, 151, 191, 193, 199, 223, 263, 271, 281, 337, 359, 401, 439, 457, 503, 521, 569, 577, 599, 641, 719, 727, 751, 839, 857, 863, 881, 887, 929, 991, 1009, 1033, 1097, 1103, 1151, 1193, 1217, 1231, 1279, 1297, 1303

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 8 divides q^2+7.

Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+7)/8]&]

lista(nn) = {forprime(p=2, nn, iferr(if (isprime(q=(p^2+7)/8), print1(q, ", ")), E, ););} \\ Michel Marcus, Feb 18 2018

A118939 Primes p such that (p^2+3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 67, 83, 101, 109, 139, 151, 157, 179, 181, 199, 211, 223, 239, 263, 277, 283, 307, 311, 337, 347, 353, 379, 389, 419, 431, 463, 491, 557, 577, 587, 619, 659, 673, 739, 757, 797, 809, 811, 829, 853, 907, 911, 953, 991, 1051

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2+3. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118940, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+3)/4]&]

A165635 Primes of the form (p^2 - 3)/2 where p is also prime.

Original entry on oeis.org

3, 11, 23, 59, 83, 179, 263, 419, 479, 683, 839, 1103, 2243, 2663, 3119, 4703, 5099, 5303, 5939, 11399, 12323, 19403, 22259, 25763, 27143, 28559, 33023, 34583, 42923, 47123, 54779, 56783, 60899, 62303, 64439, 67343, 75659, 78803, 83639, 98123

Offset: 1

Vincenzo Librandi, Sep 23 2009

The sequence could be generated by searching for squared primes p^2 in A153238.

			The prime 3=(3^2-3)/2 is generated by p=3. The prime 11=(5^2-3)/2 is generated by p=5. The prime 23 by p=7.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A110589, A153238.

[a: p in PrimesInInterval(1, 500) | IsPrime(a) where a is (p^2 - 3) div 2]; // Vincenzo Librandi, Oct 12 2012

Select[Table[(p^2 - 3)/2, {p, Prime[Range[300]]}], PrimeQ] (* Vincenzo Librandi, Oct 12 2012 *)

a(n) = (A110589(n)^2-3)/2 .

More terms from Max Alekseyev, Sep 25 2009

Comment clarified by R. J. Mathar, Oct 07 2009

A118941 Primes p such that (p^2-5)/4 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 53, 61, 71, 79, 83, 89, 97, 101, 107, 109, 113, 131, 137, 167, 173, 179, 193, 229, 241, 251, 263, 269, 277, 281, 283, 307, 311, 317, 349, 353, 373, 383, 419, 431, 439, 461, 463, 467, 563, 571, 577, 593, 607, 613, 619, 647

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2-5. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2-5)/4]&]

A118942 Primes p such that (p^2-13)/12 is prime.

Original entry on oeis.org

7, 13, 17, 19, 23, 31, 37, 41, 53, 67, 71, 73, 89, 103, 107, 113, 131, 139, 157, 163, 181, 199, 211, 233, 239, 257, 269, 283, 307, 311, 337, 359, 373, 379, 401, 419, 463, 487, 491, 499, 509, 521, 577, 593, 607, 617, 631, 647, 653, 683, 701, 733, 761, 769, 787

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 12 divides q^2-13. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118941.

Select[Prime[Range[200]],PrimeQ[(#^2-13)/12]&]

A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 25, 29, 65, 79, 101, 205, 209, 221, 245, 275, 289, 299, 349, 371, 409, 415, 449, 521, 535, 569, 571, 575, 595, 649, 661, 695, 739, 781, 791, 935, 949, 991, 1081, 1091, 1099, 1129, 1181, 1225, 1241, 1285, 1345, 1349, 1459, 1489, 1531, 1541, 1615

Offset: 1

Giuseppe Coppoletta, Mar 27 2017

nonn

All terms are obviously odd.

			25 is a term because (25^2 - 3)/2 = 311 and (25^2 + 1)/2 = 313 are twin primes.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002731, A109358, A048161, A110589, A284034.

filter:= n -> isprime((n^2-3)/2) and isprime((n^2+1)/2):
select(filter, [seq(i,i=1..2000,2)]); # Robert Israel, Apr 24 2017

Select[Range[1, 1285, 2], Times @@ Boole@ Map[PrimeQ, (#^2 + {-3, 1})/2] == 1 &] (* Michael De Vlieger, Mar 28 2017 *)

isok(n) = isprime((n^2 - 3)/2) && isprime((n^2 + 1)/2); \\ Michel Marcus, Apr 04 2017

from sympy import isprime
print([n for n in range(3, 1700, 2) if isprime((n**2 - 3)//2) and isprime((n**2 + 1)//2)]) # Indranil Ghosh, Apr 04 2017

[n for n in range(3,1700,2) if is_prime((n^2 - 3)//2) and is_prime((n^2 + 1)//2)]

A165637 Primes p such that (p^2-3)/2 is not a prime number.

Original entry on oeis.org

2, 17, 43, 53, 59, 61, 71, 83, 89, 107, 113, 127, 131, 137, 139, 149, 163, 167, 173, 179, 181, 191, 193, 199, 223, 229, 241, 251, 269, 271, 277, 281, 283, 311, 313, 317, 347, 373, 379, 383, 401, 419, 421, 431, 433, 439, 457, 461, 467, 479, 499, 503, 523, 541, 557, 563

Offset: 1

Vincenzo Librandi, Sep 23 2009

			For p=17=a(2), (17^2-3)/2=143 is not a prime. For p=43=a(3), (43^2-3)/2=923 is not a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[p: p in PrimesUpTo(600)| not IsPrime((p^2-3) div 2)]; // Vincenzo Librandi, Sep 12 2013

Select[Prime[Range[200]], !PrimeQ[(#^2 - 3) / 2]&] (* Harvey P. Dale, Dec 09 2011 *)

A000040 \ A110589. - R. J. Mathar, Sep 26 2009

Extended by R. J. Mathar, Sep 26 2009

cp's OEIS Frontend

A118940 Primes p such that (p^2+7)/8 is prime.

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A118939 Primes p such that (p^2+3)/4 is prime.

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A165635 Primes of the form (p^2 - 3)/2 where p is also prime.

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A118941 Primes p such that (p^2-5)/4 is prime.

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A118942 Primes p such that (p^2-13)/12 is prime.

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A284036 Positive integers n such that (n^2 - 3)/2 and (n^2 + 1)/2 are twin primes.

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A165637 Primes p such that (p^2-3)/2 is not a prime number.

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