A181968 a(n) = 54n^3 - 1.

53, 431, 1457, 3455, 6749, 11663, 18521, 27647, 39365, 53999, 71873, 93311, 118637, 148175, 182249, 221183, 265301, 314927, 370385, 431999, 500093, 574991, 657017, 746495, 843749, 949103, 1062881, 1185407, 1317005, 1457999, 1608713, 1769471, 1940597, 2122415

Offset: 1

Author

Arkadiusz Wesolowski, Apr 06 2012

Keywords

• easy
• nonn

Comments

a(n) is coprime to 27*n^3*(27*n^3 - 1) - 2 = A016767(n)*(A016767(n)-1) - 2.

x^3 + y^3 + z^3 = w^3 has infinitely many solutions, where every pair of elements x, y and z are coprime.

This follows from the identity a(n)^3 + (A016767(n)+1)^3 + (A016768(n)-A008588(n))^3 = (A016768(n)+A008585(n))^3 for n >= 1.

References

• Wacław Sierpiński, Czym sie zajmuje teoria liczb. Warsaw: PW "Wiedza Powszechna", 1957, pp. 59-60.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A008585, A016767.

Programs

Magma

[ 54*n^3-1 : n in [1..34]];

Maple

seq(54*n^3-1, n=1..34);

Mathematica

Table[54*n^3 - 1, {n, 34}]

PARI

vector(34, n, 54*n^3-1)

Formula

For n >= 1, a(n) = 54*A000578(n) - 1 = 2*A016767(n) - 1.

G.f.: (-1 + 57*x + 213*x^2 + 55*x^3)/(1 - x)^4.