A103558 Semiprimes of the form p^2 + q^2, where p and q are primes.

34, 58, 74, 146, 178, 194, 218, 298, 314, 365, 386, 458, 482, 533, 538, 554, 698, 818, 866, 965, 1082, 1202, 1322, 1418, 1538, 1658, 1685, 1706, 1853, 1858, 1874, 2018, 2042, 2138, 2218, 2234, 2258, 2498, 2642, 2813, 2818, 2858, 2978, 3098, 3218, 3338

Offset: 1

Author

Giovanni Teofilatto, Mar 23 2005

Keywords

• easy
• nonn

Comments

p and q must be distinct, otherwise p^2 + q^2 = 2*p*p has three prime factors. - Klaus Brockhaus

Even terms are 2*A103739. - Robert Israel, Nov 03 2017

Examples

			34 is a term because 3^2 + 5^2 = 34 = 2*17; 58 is a term because 3^2 + 7^2 = 58 = 2*29; 74 is a term because 5^2 + 7^2 = 74 = 2*37.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001358, A006881, A103739.

Programs

Maple

N:= 10000: # to get all terms <= N
P:= select(isprime, [$1..floor(sqrt(N))]):
Res:= NULL:
for i from 1 to nops(P) do
  for j from 1 to i-1 do
    r:= P[i]^2 + P[j]^2;
    if r > N then break fi;
    if numtheory:-bigomega(r) = 2 then Res:= Res, r fi;
od od:
sort(convert({Res},list)); # Robert Israel, Nov 03 2017

Mathematica

fQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; Select[ Sort[ Flatten[ Table[ Prime[p]^2 + Prime[q]^2, {p, 16}, {q, p - 1}]]], fQ[ # ] &] (* Robert G. Wilson v, Mar 23 2005 *)

PARI

{m=53;v=[];forprime(p=2,m, forprime(q=nextprime(p+1),m,if(bigomega(k=p^2+q^2)==2, v=concat(v,k))));v=vecsort(v);stop=nextprime(m+1)^2;for(j=1,length(v),if(v[j]Klaus Brockhaus

Extensions

More terms from Klaus Brockhaus and Robert G. Wilson v, Mar 23 2005