A348169 Positive integers which can be represented as A*(x^2 + y^2 + z^2) = B*(x*y + x*z + y*z) with positive integers x, y, z, A, B and gcd(A,B)=1.

3, 12, 18, 27, 30, 42, 48, 72, 75, 77, 98, 108, 120, 147, 154, 162, 168, 192, 243, 255, 260, 264, 270, 272, 273, 285, 288, 297, 300, 308, 338, 363, 378, 392, 432, 450, 462, 480, 490, 494, 507, 510, 513, 588, 616, 630, 648, 672, 675, 693, 702, 714, 722, 750, 754, 768, 798

Offset: 1

Author

Alexander Kritov, Oct 04 2021

Keywords

• nonn

Comments

The sequence represents a generalization of cases A033428 (k=1), A347960 (k=2), A347969 (k=5) with all possible k given by A331605. Instead of integer k, it utilizes the ratio B/A.

Examples

			a(6)=42: the quintuple (x,y,z) A,B is 1,2,4 (2,3) because 42 = 2*(1^2 + 2^2 + 4^2) = 3*(1*4 + 1*2 + 2*4).
  a(n)    (x,y,z)     A,  B
    3     (1,1,1)     1,  1
   12     (2,2,2)     1,  1
   18     (1,1,4)     1,  2
   27     (3,3,3)     1,  1
   30     (1,1,2)     5,  6
   42     (1,2,4)     2,  3
   48     (4,4,4)     1,  1
   72     (1,2,2)     8,  9  [also (2,2,8) 1, 2]
   75     (5,5,5)     1,  1
   77     (1,1,3)     7, 11
   98     (1,4,9)     1,  2
  108     (6,6,6)     1,  1
  120     (2,2,4)     5,  6
  147     (7,7,7)     1,  1
  154     (1,2,3)    11, 14
  162     (3,3,12)    1,  2
  168     (2,4,8)     2,  3
  192     (8,8,8)     1,  1
  243     (9,9,9)     1,  1
  255     (1,1,7)     5, 17
  260     (2,5,6)     4,  5
  264     (1,4,4)     8, 11
  270     (2,5,5)     5,  6
  272     (2,2,3)    16, 17
  288     (4,4,2)     8,  9  [also (4,4,16) 1, 2]
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Alexander Kritov, Source code

Crossrefs

Cf. A331605, A347969, A347960.

The sequence contains A033428 (A=B=1), A347969 (B=2*A), A347960 (B=5*A).

Programs

C

/* See links */

Python

from itertools import islice, count
from math import gcd
from sympy import divisors, integer_nthroot
def A348169(): # generator of terms
    for n in count(1):
        for d in divisors(n,generator=False):
            x, x2 = 1, 1
            while 3*x2 <= d:
                y, y2 = x, x2
                z2 = d-x2-y2
                while z2 >= y2:
                    z, w = integer_nthroot(z2,2)
                    if w:
                        A = n//d
                        B, u = divmod(n,x*(y+z)+y*z)
                        if u == 0 and gcd(A,B) == 1:
                            yield n
                            break
                    y += 1
                    y2 += 2*y-1
                    z2 -= 2*y-1
                else:
                    x += 1
                    x2 += 2*x-1
                    continue
                break
            else:
                continue
            break
A348169_list = list(islice(A348169(),57)) # Chai Wah Wu, Nov 26 2021