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

Offset: 1

Author

Carlos Alves, Aug 05 2009

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A050787, A050791.

Programs

Magma

[6*n^3+1 : n in [1..50]]; // Wesley Ivan Hurt, Jan 09 2017

Maple

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

Mathematica

6*Range[40]^3+1 (* or *) LinearRecurrence[{4,-6,4,-1},{7,49,163,385},40] (* Harvey P. Dale, Dec 12 2011 *)

PARI

a(n)=6*n^3+1 \\ Charles R Greathouse IV, Nov 29 2014

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]