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

1,1

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

Links

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

Example

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

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

Prog

(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

Crossrefs

Intersection of A143850 and A000040.

Cf. A001248, A002313.

Sequence in context: A266965 A081985 A087937 * A255871 A196668 A096785

Adjacent sequences: A103736 A103737 A103738 * A103740 A103741 A103742

Keyword

easy,nonn

Author

Giovanni Teofilatto, Mar 28 2005

Extensions

Corrected and extended by Walter Nissen, Jul 19 2005

Status

approved