A331605 -id:A331605 - cp's OEIS frontend

A347360 Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.

18, 72, 98, 162, 288, 338, 392, 450, 648, 722, 882, 1152, 1352, 1458, 1568, 1800, 1922, 2178, 2450, 2592, 2738, 2888, 3042, 3528, 3698, 4050, 4608, 4802, 5202, 5408, 5832, 6272, 6498, 7200, 7442, 7688, 7938, 8450, 8712, 8978, 9522, 9800, 10368, 10658, 10952, 11250, 11552, 11858

Offset: 1

Alexander Kritov, Sep 22 2021

nonn

Integers that can be represented as the sum of three squares of integers x, y, z, and additionally also satisfy x^2+y^2+z^2 = k *(x*y+ x*z + y*z), with k=2.

All possible k are given by A331605.

			For example, the third term (1,4,9) is 1^2+4^2+9^2 = 2*(1*4+1*9+4*9) = 98.
The sequence is given by
   a(n)    (x, y, z)
    18     (1,1,4)
    72     (2,2,8)
    98     (1,4,9)
   162     (3,3,12)
   288     (4,4,16)
   338     (1,9,16)
   392     (2,8,18)
   450     (5,5,20)
   648     (6,6,24)
   722     (4,9,25)
   882     (1,16,25) (3,12,27)  (7,7,28)
  1152     (8,8,32)  (2,18,32)
  1352     (2,18,32)
  1458     (9,9,36)
  1568     (4,16,36)
  1800     (10,10,40)
  1922     (1,25,36)
  2178     (11,11,44)
  2450     (5,20,45)
  2592     (12,12,48)
  2738     (9,16,49)
  2888     (8,18,50)
  3042     (3,27,48) (4,25,49) (13,13,52)
  3528     (2,32,50) (6,24,54)

E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.

Subsequence of A000378. Complement of A004215.

Cf. A033428 (case k=1), A324929, A331605 (k-numbers).

q[n_] := (s = Select[PowersRepresentations[n,3,2], AllTrue[#, #1 > 0 &]&]) != {} && MemberQ[(#[[1]]*#[[2]] + #[[2]]*#[[3]] + #[[3]]*#[[1]])& /@ s, n/2]; Select[Range[2, 12000, 2], q] (* Amiram Eldar, Oct 03 2021 *)

Empirically, such numbers appear to be a(n) = 2*b_n^2 where b_n are numbers whose product of prime indices is even (A324929).The triplet (x,y,x) is always (n*k^2, n*m^2, n*p^2).

A347969 Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.

Original entry on oeis.org

1715, 6860, 12635, 15435, 27440, 42875, 47915, 50540, 53235, 61740, 84035, 109760, 113715, 138915, 171500, 191660, 202160, 207515, 212940, 218435, 246960, 289835, 302715, 315875, 329315, 336140, 385875, 415835, 431235, 439040, 454860, 479115, 495635, 555660, 582435, 619115, 686000

Offset: 1

Alexander Kritov, Sep 23 2021

nonn

The general problem is to find such numbers which can be represented as the sum of three squares of integers x, y, z, and additionally also satisfy: x^2 + y^2 + z^2 = k*(x*y + x*z + y*z).
For k=1 it is simply a(n) = 3*n^2 given by A033428.
For k=2 it is A347360.
The present sequence is for the next k=5.
All possible k-numbers are listed by A331605.

			    a(n)      ( x,  y,   z)
  ------      -------------
    1715      ( 3,  5,  41)
    6860      ( 6, 10,  82)
   12635      ( 5, 17, 111)
   15435      ( 9, 15, 123)
   27440      (12, 20, 164)
   42875      (15, 25, 205)
   47915      ( 3, 41, 215)
   50540      (10, 34, 222)
   53235      ( 5, 41, 227)
   61740      (18, 30, 246)
   84035      (21, 35, 287)
  109760      (24, 40, 328)

E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985.

Cf. A000378, A033428, A331605 (all possible k-numbers), A347360.

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

Original entry on oeis.org

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

Alexander Kritov, Oct 04 2021

nonn

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.

			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]

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

Cf. A331605, A347969, A347960.

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

C
```
/* See links */
```

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

cp's OEIS Frontend

A347360 Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.

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A347969 Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.

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

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cp's OEIS Frontend

A347360 Numbers that can be represented as the sum of squares of 3 numbers and also equal to twice the sum of their joint products.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula

A347969 Numbers which are sum of three squares of positive numbers and also 5 times of the sum of their joint products.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Python

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