A271576 -id:A271576 - cp's OEIS frontend

A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

10, 17, 45, 80, 123, 136, 225, 234, 260, 270, 291, 325, 360, 365, 459, 510, 514, 640, 666, 745, 984, 1025, 1088, 1215, 1225, 1250, 1305, 1450, 1466, 1565, 1740, 1753, 1800, 1872, 1950, 1970, 2022, 2080, 2125, 2160, 2328, 2600, 2628, 2880, 2920, 3172, 3185

Offset: 1

XU Pingya, Oct 14 2017

nonn

Let i, j and k be nonnegative integers, m > n be positive integers. As ((m^2 - n^2)^(3*i+1) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k))^2 + ((m^2 - n^2)^i * (2*m*n)^(j+1) * (m^2 + n^2)^k)^6 = ((m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1))^2, so that the number of the form (m^2 - n^2)^(3*i) * (2*m*n)^(3*j+2) * (m^2 + n^2)^(3*k+1) is a term.
When (x, y, z) is a solution of x^2 + y^4 = z^2 (i.e., z = A271576(n)), (x^(3*i+1) * y^(3*j+1) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j+1) * z^(3*k+1)) is a solution of x^2 + y^6 = z^2.
When (x, y, z) is a solution of x^2 + y^6 = z^2, (x^(3*i+1) * y^(3*j) * z^(3*k), x^i * y^(j+1) * z^k, x^(3*i) * y^(3*j) * z^(3*k+1)) is also a solution of x^2 + y^6 = z^2.

			6^2 + 2^6 = 10^2, 10 is a term.
15^2 + 2^6 = 17^2, 17 is a term.

Cf. A009003, A070745, A228946, A271576, A274035, A293692, A293694.

z[n_] := Count[n^2 - Range[(n^2 - 1)^(1/6)]^6, _?(IntegerQ[Sqrt[#]] &)] > 0; Select[Range[3200], z]

A349663 Positive numbers x for which x^2 can be expressed as z^2 - y^4 with y != 0.

Original entry on oeis.org

3, 12, 15, 17, 27, 30, 40, 42, 48, 60, 63, 68, 75, 77, 90, 95, 99, 105, 108, 112, 120, 130, 135, 140, 147, 153, 156, 160, 165, 168, 192, 195, 220, 240, 243, 252, 270, 272, 273, 300, 301, 308, 312, 315, 323, 350, 360, 363, 375, 378, 380, 396, 399, 420, 425, 432, 448, 462, 480, 495, 507

Offset: 1

Karl-Heinz Hofmann and Hugo Pfoertner, Dec 09 2021

nonn

This sequence is closely related to A271576.
Conditions to be satisfied for a solution:
- z cannot be a square.
- z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
- If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and y too.
- If z is even, x and y must be even too.
- The lower limit of the ratio x/y is sqrt(2).
Multiple solutions are possible; e.g., term 420 has 5 solutions.

			The 5 solutions corresponding to a(54) = 420 are 420^2 = 176400 = 444^2 - 12^4 = 580^2 - 20^4 = 609^2 - 21^4 = 1295^2 - 35^4 = 3164^2 - 56^4.

Cf. A000290, A000583, A271576, A002144, A002145.

A349664 a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1.

Original entry on oeis.org

0, 1, 2, 3, 2, 7, 2, 5, 4, 7, 2, 17, 2, 7, 12, 7, 2, 13, 2, 17, 12, 7, 2, 27, 4, 7, 6, 17, 2, 37, 2, 9, 12, 7, 12, 31, 2, 7, 12, 27, 2, 37, 2, 17, 22, 7, 2, 37, 4, 13, 12, 17, 2, 19, 12, 27, 12, 7, 2, 87, 2, 7, 22, 11, 12, 37, 2, 17, 12, 37, 2, 49, 2, 7, 22

Offset: 1

Karl-Heinz Hofmann, Dec 13 2021

nonn

If n is an odd prime^i, the number of solutions is 2*i.
If n = 2^i, the number of solutions is 2*i-1.
These two facts are not generally valid in reverse for terms > 6.
If a(n) = 2, n is an odd prime. This is generally valid in reverse.
For more information about these facts see the link.
The calculation of the terms is done with an algorithm of Jon E. Schoenfield, which is described in A349324.
Conditions to be satisfied for a valid, countable solution:
- z cannot be a square.
- z must have at least one prime factor of the form p == 1 (mod 4), a Pythagorean prime (A002144).
- If z has prime factors of the form p == 3 (mod 4), which are in A002145, then they must appear in the prime divisor sets of x and n too.
- If z is even, x and n must be even too.
- The lower limit of the ratio x/n is sqrt(2).
- high limits of z and x:
           |    n is odd      |    n is even
  ---------+------------------+------------------
  z limit  | (n^4 + 1)/2      |  (n^4 + 4)/4
  x limit  | (n^4 + 1)/2 - 1  |  (n^4 + 4)/4 - 2

			a(6) = 7 (solutions): 6^4 = 1296 = 325^2 - 323^2 = 164^2 - 160^2 = 111^2 - 105^2 = 85^2 - 77^2 = 60^2 - 48^2 = 45^2 - 27^2 = 39^2 - 15^2.

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000
Karl-Heinz Hofmann, What the terms can tell about n.

Cf. A000290, A000583, A271576, A349663, A002144, A002145, A346115.

Cf. A345645, A345700, A345968, A346110, A348655, A349324.

a[n_] := Length[Solve[n^4 == z^2 - x^2 && x >= 1 && z >= 1, {x, z}, Integers]]; Array[a, 75] (* Amiram Eldar, Dec 14 2021 *)

a(n) = numdiv(if(n%2, n^4, n^4/4))\2; \\ Jinyuan Wang, Dec 19 2021

A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

Original entry on oeis.org

5, 15, 20, 34, 39, 41, 45, 55, 60, 65, 80, 85, 111, 125, 135, 136, 145, 150, 156, 164, 175, 180, 194, 219, 220, 240, 245, 255, 260, 265, 299, 306, 313, 320, 325, 340, 351, 353, 369, 371, 375, 405, 410, 444, 445, 455, 495, 500, 505, 514, 525, 540, 544

Offset: 1

Altug Alkan, Jun 03 2016

nonn

A271576 is a subsequence.
Terms that are not in A271576 are 55, 220, 299, ...
Sequence is infinite since if k is a term then also t^2*k is a term, for every t>0. - Giovanni Resta, Jun 04 2016

			5 is a term because 75^2 + 10^4 = 5^6.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A111925, A266212, A271576.

q[n_] := {} != Select[Range[n^(1/4)]^4, n > # && IntegerQ@ Sqrt[n - #] &]; Select[ Range[100], q[#^6] &] (* Giovanni Resta, Jun 04 2016 *)

A349665 Record terms of A349664.

Original entry on oeis.org

0, 1, 2, 3, 7, 17, 27, 37, 87, 137, 157, 187, 247, 437, 687, 787, 937, 1237, 2187, 3437, 3937, 4687, 6187, 8437, 10937, 17187, 19687, 23437, 30937, 42187, 54687, 55687, 85937, 98437, 117187, 154687, 210937, 223437, 273437, 278437, 304687, 429687, 492187, 585937

Offset: 1

Karl-Heinz Hofmann, Dec 18 2021

nonn

Terms are the record numbers of solutions for the equation: y^4 = z^2 - x^2.

			Number of | y | Factorization
solutions |   |      of y
----------+---+--------------
        0 | 1 |  -
        1 | 2 | [2]
        2 | 3 | [3]
        3 | 4 | [2, 2]
        7 | 6 | [2, 3]
        :   :      :
For more terms with y and factorization of y see link.

Karl-Heinz Hofmann, Full table up to y <= 10^12

Cf. A000290, A000583, A002144, A002145, A271576, A346115, A349663.

lista(nn) = my(f, r); print1("0, 1, 2"); forstep(n=4, nn, 2, f=factor(n)[, 2]; if(rJinyuan Wang, Dec 19 2021

More terms from Jinyuan Wang, Dec 19 2021

cp's OEIS Frontend

A293690 Numbers z such that x^2 + y^6 = z^2 for positive integers x and y.

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A349663 Positive numbers x for which x^2 can be expressed as z^2 - y^4 with y != 0.

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A349664 a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1.

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A273908 Integers z such that x^2 + y^4 = z^6 where x, y, z > 0, is soluble.

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A349665 Record terms of A349664.

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