

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
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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*i1.
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

KarlHeinz Hofmann, Table of n, a(n) for n = 1..10000
KarlHeinz 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

KarlHeinz Hofmann, Dec 13 2021


STATUS

approved



