A109358 -id:A109358 - cp's OEIS frontend

A110589 Primes p such that 2*q+3 = p^2, where q is prime.

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 47, 67, 73, 79, 97, 101, 103, 109, 151, 157, 197, 211, 227, 233, 239, 257, 263, 293, 307, 331, 337, 349, 353, 359, 367, 389, 397, 409, 443, 449, 463, 487, 491, 509, 521, 547, 569, 571, 587, 613, 619, 653, 661, 673, 727

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A098828, A109358.

[p: p in PrimesUpTo(1000) | IsPrime((p^2-3) div 2)]; // Vincenzo Librandi, Mar 19 2015

ispower := proc(n,b) andmap(proc(w) evalb(w[2] mod b = 0) end, ifactors(n)[2]) end: a:=2: SQRTP||a:=[]; for z from 1 to 1 do for n from 1 to 10000 do p:=ithprime(n); m:=a*p+a+1; if ispower(m,2) and isprime(sqrt(m)) then SQRTP||a:=[op(SQRTP||a),sqrt(m)] fi od; od; SQRTP||a;

fQ[n_] := PrimeQ[(n^2 - 3)/2]; Select[ Prime@ Range@129, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *)
Select[Table[Sqrt[2 Prime[n] + 3], {n, 1, 30000}], PrimeQ] (* Vincenzo Librandi, Mar 19 2015 *)

More terms from Robert G. Wilson v, Jun 19 2006

A098828 Primes of the form 2n^2 + 2n - 1.

Original entry on oeis.org

3, 11, 23, 59, 83, 179, 263, 311, 419, 479, 683, 839, 1103, 1511, 2111, 2243, 2663, 2963, 3119, 4139, 4703, 5099, 5303, 5939, 7079, 10223, 11399, 12011, 12323, 12959, 17483, 19403, 21011, 21839, 22259, 24419, 25763, 27143, 27611, 28559, 30011

Offset: 1

Giovanni Teofilatto, Oct 09 2004

nonn

a(n)==3 (mod 4).
Equivalently primes p such that 2p+3 is square.
Also 3 followed by primes p of the form 2*n^2+6*n+3 = 2*(n+2)^2-2*(n+2)-1 (see the first comment) such that 2^(p-1)+3 is not prime. - Vincenzo Librandi, Jan 03 2009; M. F. Hasler, Jan 07 2009; R. J. Mathar, Jan 14 2009; Bruno Berselli, Sep 23 2013

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

Cf. A109358, A109367, A153238

[3] cat [ p: p in PrimesUpTo(30100) | exists(t){ n: n in [1..Isqrt(p div 2)] | 2*n^2+6*n+3 eq p } and not IsPrime(2^(p-1)+3) ];

Select[Table[Prime[n], {n, 3500}], IntegerQ[(2# + 3)^(1/2)] &] (* Ray Chandler, Oct 26 2004 *)

list(lim)=my(v=List()); for(n=1,oo, my(t=2*n*(n+1)-1); if(t>lim, return(Vec(v))); if(isprime(t), listput(v,t))) \\ Charles R Greathouse IV, Feb 26 2025

a(n) = (A109367(n) - 3)/2.

Corrected by Ray Chandler, Oct 26 2004
Edited by N. J. A. Sloane, Jan 25 2009
Name edited by Charles R Greathouse IV, Feb 26 2025

A109367 Squares of the form 2*p + 3, where p is a prime.

Original entry on oeis.org

9, 25, 49, 121, 169, 361, 529, 625, 841, 961, 1369, 1681, 2209, 3025, 4225, 4489, 5329, 5929, 6241, 8281, 9409, 10201, 10609, 11881, 14161, 20449, 22801, 24025, 24649, 25921, 34969, 38809, 42025, 43681, 44521, 48841, 51529, 54289, 55225, 57121

Offset: 1

Giovanni Teofilatto, Aug 23 2005

nonn

The first seven terms are primes squared: 3^2, 5^2, 7^2, 11^2, 13^2, 17^2, 19^2, 23^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A098828, A109358.

Select[Table[(2Prime[n] + 3)^(1/2), {n, 3500}], IntegerQ]^2 (* Ray Chandler, Aug 25 2005 *)
Select[2*Prime[Range[3500]]+3,IntegerQ[Sqrt[#]]&] (* Harvey P. Dale, Sep 07 2020 *)

is(n)=issquare(n) && n%2 && isprime(n\2-1) \\ Charles R Greathouse IV, Aug 06 2013

a(n) = A109358(n)^2 = 2*A098828(n) + 3.

Extended by Ray Chandler and Robert G. Wilson v, Aug 25 2005

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)]

A110588 Squares of the form 2*p+3 that are squares of primes.

Original entry on oeis.org

9, 25, 49, 121, 169, 361, 529, 841, 961, 1369, 1681, 2209, 4489, 5329, 6241, 9409, 10201, 10609, 11881, 22801, 24649, 38809, 44521, 51529, 54289, 57121, 66049, 69169, 85849, 94249, 109561, 113569, 121801, 124609

Offset: 1

Walter Kehowski, Sep 13 2005

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A098828, A109358.

ispower := proc(n,b) andmap(proc(w) evalb(w[2] mod b = 0) end, ifactors(n)[2]) end: a:=2: PW||a:=[]; for z from 1 to 1 do for n from 1 to 1000 do p:=ithprime(n); m:=a*p+a+1; if ispower(m,2) and isprime(sqrt(m)) then PW||a:=[op(PW||a),m] fi od; od; PW||a;

isok(n) = issquare(n) && isprime(sqrtint(n)) && (type(p=(n-3)/2) == "t_INT") && isprime(p) \\ Michel Marcus, Aug 06 2013

v=List();forprime(p=2,1e4,if(isprime(p^2\2-1),listput(v,p^2))); Vec(v) \\ Charles R Greathouse IV, Aug 06 2013

More terms from Michel Marcus, Aug 06 2013

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A110589 Primes p such that 2*q+3 = p^2, where q is prime.

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A098828 Primes of the form 2*n^2 + 2*n - 1.

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A109367 Squares of the form 2*p + 3, where p is a 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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A110588 Squares of the form 2*p+3 that are squares of primes.

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Maple

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PARI

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A098828 Primes of the form 2n^2 + 2n - 1.