A103739 Primes which are half the sum of 2 squares of primes.

17, 29, 37, 73, 89, 97, 109, 149, 157, 193, 229, 241, 269, 277, 349, 409, 433, 541, 601, 661, 709, 769, 829, 853, 929, 937, 1009, 1021, 1069, 1109, 1117, 1129, 1249, 1321, 1409, 1429, 1489, 1549, 1609, 1669, 1753, 1789, 1801, 1873, 2029, 2089, 2161, 2221

Offset: 1

Author

Giovanni Teofilatto, Mar 28 2005

Keywords

• easy
• nonn

Comments

Primes of the form x^2 + y^2, where x > y > 0, such that x-y = p and x+y = q are primes. Proof: (p^2+q^2)/2 = ((x-y)^2+(x+y)^2)/2 = x^2+y^2 so we have x = (p+q)/2 and y = (q-p)/2. - Thomas Ordowski, Sep 24 2012

All terms == 1 or 5 (mod 12). - Thomas Ordowski, Jun 28 2013

Or, primes in A143850. - Zak Seidov, Jun 06 2015

Examples

			17 is in the sequence because (3^2 + 5^2) / 2 = 17.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Intersection of A143850 and A000040.

Cf. A001248, A002313.

Programs

Maple

Primes:= select(isprime,[seq(2*i+1,i=1..400)]):
Psq:= map(`^`,Primes,2):
M:= max(Psq):
S:= select(t -> t <= M/2 and isprime(t),{seq(seq((Psq[i]+Psq[j])/2, j=1..i-1),i=1..nops(Psq))}):
sort(convert(S,list)); # Robert Israel, Jun 08 2015

PARI

list(lim)=my(v=List(), p2, t); lim\=1; if(lim<9, lim=9); forprime(p=3, sqrtint(2*lim-9), p2=p^2; forprime(q=3, min(sqrtint(2*lim-p2), p), if(isprime(t=(p2+q^2)/2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Feb 14 2017

Extensions

Corrected and extended by Walter Nissen, Jul 19 2005