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A349664 a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

LINKS

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Karl-Heinz Hofmann, What the terms can tell about n.

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

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

Sequence in context: A144456 A262427 A333986 * A266258 A180916 A319374

Adjacent sequences:  A349661 A349662 A349663 * A349665 A349666 A349667

KEYWORD

nonn

AUTHOR

Karl-Heinz Hofmann, Dec 13 2021

STATUS

approved

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Last modified May 23 04:06 EDT 2022. Contains 353959 sequences. (Running on oeis4.)