A068601 a(n) = n^3 - 1.

0, 7, 26, 63, 124, 215, 342, 511, 728, 999, 1330, 1727, 2196, 2743, 3374, 4095, 4912, 5831, 6858, 7999, 9260, 10647, 12166, 13823, 15624, 17575, 19682, 21951, 24388, 26999, 29790, 32767, 35936, 39303, 42874, 46655, 50652, 54871, 59318, 63999, 68920

Offset: 1

Author

Naohiro Nomoto, Mar 28 2002

Keywords

• nonn
• easy

Comments

a(n) is the least positive integer k such that k can only contain 'n-1' in exactly 2 different bases B, where 1 < B <= k.

Apart from the first term, the same as A135300. - R. J. Mathar, Apr 29 2008

A058895(n)^3 + a(n)^3 + A033562(n)^3 = A185065(n)^3. - Vincenzo Librandi, Mar 13 2012

Numbers k such that for every nonnegative integer m, k^(3*m+1) + k^(3*m) is a cube. - Arkadiusz Wesolowski, Aug 10 2013

Examples

			For n=6; 215 written in bases 6 and 42 is 555, 55 and (555, 55) are exactly 2 different bases.
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A003215, A005448, A016921, A033562, A058895, A129294, A135300, A185065, A339604.

Programs

GAP

List([1..45],n->n^3-1); # Muniru A Asiru, Oct 23 2018

Magma

[n^3-1: n in [1..40]]; // Vincenzo Librandi, Mar 11 2012

Maple

A068601:=n->n^3-1: seq(A068601(n), n=1..50); # Wesley Ivan Hurt, Jul 23 2014

Mathematica

f[n_]:=n^3-1;f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
LinearRecurrence[{4,-6,4,-1},{0,7,26,63},50] (* Vincenzo Librandi, Mar 11 2012 *)
Range[50]^3 - 1 (* Wesley Ivan Hurt, Jul 23 2014 *)

PARI

a(n)=n^3-1

Python

for n in range(1,50): print(n**3-1, end=', ') # Stefano Spezia, Nov 21 2018

Formula

Partial sums of A003215, hex (or centered hexagonal) numbers: 3*n(n+1)+1. - Jonathan Vos Post, Mar 16 2006

G.f.: x^2*(7-2*x+x^2)/(1-x)^4. - Colin Barker, Feb 12 2012

4*a(m^2-2*m+2) = (m^2-m+1)^3 + (m^2-m-1)^3 + (m^2-3*m+3)^3 + (m^2-3*m+1)^3. - Bruno Berselli, Jun 23 2014

a(n) = Sum_{i=1..n-1} (i+1)^3 - i^3. - Wesley Ivan Hurt, Jul 23 2014

Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(3*n) - 1) = A339604. - Amiram Eldar, Nov 06 2020

Product_{n>=2} (1 + 1/a(n)) = 3*Pi*sech(sqrt(3)*Pi/2). - Amiram Eldar, Jan 20 2021

E.g.f.: 1 + exp(x)*(x^3 + 3*x^2 + x - 1). - Stefano Spezia, Jul 06 2021