A349987 -id:A349987 - cp's OEIS frontend

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

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

nonn

The only square in this sequence is 49.

			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.

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?

Contains A079705, A244146, A349987.

Subsequence of A024614.

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));

A350655 a(n) is the least positive number that can be written as p^2 + p*q + q^2 in exactly n ways where p and q are primes and p <= q.

Original entry on oeis.org

1, 12, 147, 57603, 160797, 4611243, 36822513, 878112417, 2069618187, 9891199227, 9098192883, 27885254943, 73104587283, 132014176203, 3457814397303, 1449081095007, 5644476547437, 9051074413563, 31516441411377, 8343886414773, 5272121828883

Offset: 0

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

a(9) > 5*10^9. - Jon E. Schoenfield, Jan 13 2022

			a(3) = 57603 as 57603 has the three representations 57603 = 2^2 + 2*239 + 239^2 = 31^2 + 31*223 + 223^2 = 101^2 + 101*173 + 173^2 and no smaller number has this property.

Cf. A349987.

N:= 10^9:
Q:= Vector(N,datatype=integer[4]):
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
T:= Array(0..7): T[0]:= 1:
nP:= nops(P):
for i from 1 to nP do
  for j from 1 to i do
    v:= P[i]^2 + P[i]*P[j] + P[j]^2;
    if v > N then break fi;
    Q[v]:= Q[v]+1;
    if T[Q[v]] = 0 or v < T[Q[v]] then T[Q[v]]:= v fi
od od:
convert(T,list);

a(8) from Jon E. Schoenfield, Jan 13 2022

a(9)-a(20) from Martin Ehrenstein, Jan 24 2022

cp's OEIS Frontend

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

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