A395347 Squarefree part of b^2 - b*c + c^2 for triples (a,b,c) with a<=b<=c, a+b>c, gcd(a,b,c)=1, ordered by perimeter.

1, 1, 1, 7, 1, 1, 13, 13, 1, 21, 19, 1, 1, 21, 1, 31, 7, 1, 21, 31, 1, 1, 43, 39, 37, 31, 1, 43, 39, 31, 1, 57, 13, 1, 1, 43, 39, 1, 57, 1, 1, 43, 1, 73, 67, 1, 7, 57, 61, 13, 1, 43, 1, 73, 67, 7, 57, 1, 1, 91, 21, 79, 73, 19, 67, 1, 57, 91, 1, 79

Offset

1,4

Comments

All terms are odd.

The quadratic form b^2 - b*c + c^2 generates numbers whose prime factors are 3, primes congruent to 1 mod 3, and squares of primes congruent to 2 mod 3. The squarefree part of these numbers yields numbers whose prime factors are only 3 and primes congruent to 1 mod 3 (A007645).

The sequence is sorted by the sum a+b+c (perimeter).

Links

Table of n, a(n) for n=1..70.

Example

For the triangle (1,1,1): b^2 - b*c + c^2 = 1, squarefree part is 1.
For the triangle (1,2,2): b^2 - b*c + c^2 = 4, squarefree part is 1.
For the triangle (2,2,3): b^2 - b*c + c^2 = 7, squarefree part is 7.

Prog

(Python)
from sympy.ntheory.factor_ import core
from math import gcd
def A395347(terms=70):
    seq, p = [], 3
    while len(seq) < terms:
        for a in range(1, p // 3 + 1):
            for b in range(a, (p - a) // 2 + 1):
                c = p - a - b
                if c >= b and a + b > c and gcd(a, b, c) == 1:
                    seq.append(core(b * b - b * c + c * c, 2))
        p += 1
    return seq[:terms]

Crossrefs

Cf. A007645, A007913, A003136, A034017, A034021, A266836.

Sequence in context: A287326 A131065 A081580 * A082110 A275526 A141597

Adjacent sequences: A395344 A395345 A395346 * A395348 A395349 A395350

Keyword

nonn,easy

Author

Chen Zhao, Apr 20 2026

Status

approved