A110284 Squares of the form 4p - 3, where p is a prime.

9, 25, 49, 121, 169, 289, 625, 841, 961, 1225, 1681, 1849, 2401, 3025, 4489, 5929, 6889, 10201, 11881, 13225, 14161, 15625, 17689, 19321, 20449, 22801, 24025, 24649, 25921, 32041, 32761, 39601, 41209, 44521, 48841, 49729, 55225, 57121, 69169

Offset: 1

Author

Giovanni Teofilatto, Sep 07 2005

Keywords

• nonn

Comments

Also: squares of the form 2*s-3, where s is a semiprime, A107317. - Franklin T. Adams-Watters, Jun 28 2010

Squares are less dense then primes and easy to generate so it's faster to check squares if they are of the required form than to check if primes are of the required form. - David A. Corneth, Oct 15 2018

Links

• David A. Corneth, Table of n, a(n) for n = 1..16289 (First 1316 terms by Marius A. Burtea, terms < 10^11)

Crossrefs

Cf. A002383, A002384, A088503, A001358.

Programs

Magma

[4*p - 3: p in PrimesUpTo(10^5)|IsSquare (4*p - 3)]; // Vincenzo Librandi, Oct 17 2018

Mathematica

Select[ 4Prime[ Range[2000]] - 3, IntegerQ[ Sqrt[ # ]] &] (* Robert G. Wilson v, Sep 20 2005 *)

PARI

isok(n) = issquare(n) && (p=(n+3)/4) && (frac(p)==0) && isprime(p); \\ Michel Marcus, Oct 15 2018

PARI

upto(n) = my(res = List()); forstep(i = 3, sqrtint(n), 2, if(isprime((i^2+3)/4), listput(res, i^2))); res \\ David A. Corneth, Oct 15 2018

Formula

a(n) = 4*A002383(n) - 3 = A088503(n-1)^2.

Extensions

Extended by Ray Chandler, Sep 07 2005