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A349664
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a(n) is the number of solutions for n^4 = z^2 - x^2 with {z,x} >= 1.
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2
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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
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1,3
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COMMENTS
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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.
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
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n) = numdiv(if(n%2, n^4, n^4/4))\2; \\ Jinyuan Wang, Dec 19 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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