A163827 - OEIS

A163827

a(n) = 6n^3 + 1, solution z in Diophantine equation x^3 + y^3 = z^3 - 2. It may be considered a Fermat near miss by 2.

7, 49, 163, 385, 751, 1297, 2059, 3073, 4375, 6001, 7987, 10369, 13183, 16465, 20251, 24577, 29479, 34993, 41155, 48001, 55567, 63889, 73003, 82945, 93751, 105457, 118099, 131713, 146335, 162001, 178747, 196609, 215623, 235825, 257251, 279937

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OFFSET

1,1

COMMENTS

It is easy to check that with x = 6n^2, y = 6n^3 - 1, and this z = 6n^3 + 1, it satisfies the Diophantine equation x^3 + y^3 = z^3 - 2. Thus these are near-misses for Fermat equation.

For n>2, it seems to be the only solution of x^n + y^n = z^n - 2 (or even that differ by 2 from FLT, see A050787 and A050791 for solutions that differ by 1). As 2 is not a cube, these solutions are not included in the theory for x^3 + y^3 = u^3 + v^3.

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

FORMULA

a(n) = 6n^3+1.

a(1)=7, a(2)=49, a(3)=163, a(4)=385, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). [Harvey P. Dale, Dec 12 2011]

G.f.: (-x^3+9*x^2+21*x+7)/(x-1)^4. [Harvey P. Dale, Dec 12 2011]

EXAMPLE

For n=1, a(1)=7 and 7^3 - 2 (=341) = 5^3 + 6^3.

For n=2, a(2)=49 and 49^3 - 2 (=117647) = 24^3 + 47^3.

MAPLE

A163827:=n->6*n^3+1: seq(A163827(n), n=1..50); # Wesley Ivan Hurt, Jan 09 2017