Alexander Kritov - cp's OEIS frontend

A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

1, 5, 9, 13, 17, 29, 33, 37, 41, 45, 57, 61, 65, 77, 81, 93, 97, 109, 113, 117, 129, 133, 137, 141, 145, 165, 177, 181, 185, 197, 209, 213, 217, 221, 233, 245, 249, 261, 265, 277, 289, 301, 305, 309, 313, 325, 329, 333, 337, 341, 361, 373, 385, 397, 401, 413, 417, 421, 433, 437, 449

Offset: 0

Alexander Kritov, Nov 21 2021

nonn

Consider a 2D lattice, where the Cartesian coordinates x and y are legs of the Pythagorean triangle. Thus the notion of Pythagorean triple is extended to the cases when sides x, y are in Z (i.e., sides also include negative integers and zero). The sequence gives the number of such triples on or inside a circle of radius n.

Partial sums of A046109.

			Sides (coordinates)                                                       a(n)
------------------------------------------------------------------------------
(0,0)                                                                       1
(-1,0)(0,-1)(0,1)(1,0)                                                      5
(-2,0)(0,-2)(0,2)(2,0)                                                      9
(-3,0)(0,-3)(0,3)(3,0)                                                     13
(-4,0)(0,-4)(0,4)(4,0)                                                     17
(-5,0)(-4,-3)(-4,3)(-3,-4)(-3,4)(0,-5)(0,5)(3,-4)(3,4)(4,-3)(4,3)(5,0)     29
(-6,0)(0,-6)(0,6)(6,0)                                                     33
(-7,0)(0,-7)(0,7)(7,0)                                                     37
(-8,0)(0,-8)(0,8)(8,0)                                                     41
(-9,0)(0,-9)(0,9)(9,0)                                                     45
(-10,0)(-8,-6)(-8,6)(-6,-8)(-6,8)(0,-10)(0,10)(6,-8)(6,8)(8,-6)(8,6)(10,0) 57
(-11,0)(0,-11)(0,11)(11,0)                                                 61
(-12,0)(0,-12)(0,12)(12,0)                                                 65

Alexander Kritov, Source code

Cf. A046080, A211432, A046109 (first differences), A349536 (in 1/8 sector).

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

f(n) = if(n==0, return(1)); my(f=factor(n)); 4*prod(i=1, #f~, if(f[i, 1]%4==1, 2*f[i, 2]+1, 1)); \\ A046109
a(n) = sum(k=0, n, f(k)); \\ Michel Marcus, Nov 27 2021

a(n) = (A211432(n) + 1)/2.

a(n) = a(n-1) + 4 + 8*A046080(n).

A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 75, 76, 77, 78, 79, 80, 84, 85, 86, 87, 89

Offset: 1

Alexander Kritov, Nov 21 2021

nonn

Number of Pythagorean triples with hypotenuse less than or equal to the next one.

			The count of non-primitive Pythagorean triples as they appear in order of increasing hypotenuse:
.
       Hypotenuse
   n  (A009003(n))       Sides       a(n)
  --  ------------  ---------------  ----
   1        5            (3,4)         1
   2       10            (6,8)         2
   3       13            (5,12)        3
   4       15            (9,12)        4
   5       17            (8,15)        5
   6       20           (12,16)        6
   7       25       (7,24), (15,20)    8
   8       26           (10,24)        9
   9       29           (20,21)       10

W. Sierpinski, Pythagorean Triangles, Dover Publications, 2003.

Alexander Kritov, Table of n, a(n) for n = 1..1050
Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Alexander Kritov, C code that generates b-file
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A008846, A009003.
Cf. A020882, A081804, A008846, A020883, A046086.
Cf. A349538 (extension to the full circle of Z^2 lattice).

// see enclosed main.c
for (long j=1;j< 101;++j)
{
for (long k=1;k< 101;++k)
{
if (k<=j)   // to avoid pairs (as we need 1/8 or quarter plane)
    {
          double hyp=sqrt(j*j+k*k);
          double c= (double) floor (hyp );
if   (fabs(hyp - c) < DBL_EPSILON)  arr[r++]= (long) c;
}}}
bubbleSort(arr, r);//sort by hypotenuse increase
for (long j=0;j< r;++j)
{
   if  ( arr[j] != arr[j+1] )
    {
        // write to file: j is the sequence value a[n]*2
        // arr[j] is the hypotenuse value
    }
}

Conjecture: the increment is a(n+1) - a(n) = 2^(m-1), where m is the sum of all powers of the Pythagorean primes (A002144) in the factorization of hypotenuse h(n+1) (see Eckert for PPT). However, starting from 58 the increment is 3.

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

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.

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.

Original entry on oeis.org

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).

cp's OEIS Frontend

User: Alexander Kritov

A349538 The number of pseudo-Pythagorean triples (which allow negative or 0 sides) on a 2D lattice that are on or inside a circle of radius n.

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A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

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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.

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Python

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

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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Crossrefs

Programs

Mathematica

Formula

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.