A349986 Numbers that can be represented as p^2 + p*q + q^2 where p and q are primes.

12, 19, 27, 39, 49, 67, 75, 79, 109, 147, 163, 199, 201, 217, 247, 259, 309, 327, 349, 363, 399, 403, 427, 433, 457, 481, 507, 543, 579, 597, 607, 669, 679, 691, 739, 777, 867, 903, 937, 973, 997, 1011, 1027, 1063, 1083, 1093, 1141, 1209, 1227, 1281, 1327, 1387, 1423, 1447, 1489, 1533, 1579, 1587

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 09 2022

Keywords

• nonn

Comments

The only square in this sequence is 49.

Examples

			a(3) = 27 is a term because 27 = 3^2+3*3+3^2.
a(4) = 39 is a term because 39 = 2^2+2*5+5^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Mathematics StackExchange, Largest square written as p^2+pq+q^2 where p, q are primes?

Crossrefs

Contains A079705, A244146, A349987.

Subsequence of A024614.

Programs

Maple

N:= 10^4: # for terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
nP:= nops(P):
S:= {}:
for i from 1 to nP do
  for j from 1 to i do
    x:= P[i]^2 + P[i]*P[j]+P[j]^2;
    if x > N then break fi;
    S:= S union {x};
od od:
sort(convert(S,list));