A045637 Primes of the form p^2 + 4, where p is prime.

13, 29, 53, 173, 293, 1373, 2213, 4493, 5333, 9413, 10613, 18773, 26573, 27893, 37253, 54293, 76733, 85853, 94253, 97973, 100493, 120413, 139133, 214373, 237173, 253013, 299213, 332933, 351653, 368453, 375773, 458333, 552053, 619373

Offset: 1

Author

Felice Russo

Keywords

• nonn
• easy

Comments

These are the only primes that are the sum of two primes squared. 11 = 3^2 + 2 is the only prime of the form p^2 + 2 because all primes greater than 3 can be written as p=6n-1 or p=6n+1, which allows p^2+2 to be factored. - T. D. Noe, May 18 2007

Infinite under the Bunyakovsky conjecture. - Charles R Greathouse IV, Jul 04 2011

All terms > 29 are congruent to 53 mod 120. - Zak Seidov, Nov 06 2013

Examples

			29 belongs to the sequence because it equals 5^2 + 4.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Yang Ji, Several special cases of a square problem, arXiv:2105.05250 [math.GM], 2021.

Crossrefs

The corresponding primes p are in A062324.

Subsequence of A005473 (and thus A185086).

Cf. A094473-A094479.

Programs

Mathematica

Select[Prime[ # ]^2+4&/@Range[140], PrimeQ]

PARI

forprime(p=2,1e4,if(isprime(t=p^2+4),print1(t","))) \\ Charles R Greathouse IV, Jul 04 2011

Formula

a(n) = A062324(n)^2 + 4. - Zak Seidov, Nov 06 2013

Extensions

Edited by Dean Hickerson, Dec 10 2002