A055040 Numbers of the form 3^(2i+1)*(3*j+2).

6, 15, 24, 33, 42, 51, 54, 60, 69, 78, 87, 96, 105, 114, 123, 132, 135, 141, 150, 159, 168, 177, 186, 195, 204, 213, 216, 222, 231, 240, 249, 258, 267, 276, 285, 294, 297, 303, 312, 321, 330, 339, 348, 357, 366, 375, 378, 384, 393, 402, 411

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn
• nice

Comments

Numbers not of the form x^2+y^2+3z^2.

Numbers whose squarefree part is congruent to 6 modulo 9. - Peter Munn, May 17 2020

The asymptotic density of this sequence is 1/8. - Amiram Eldar, Mar 08 2021

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Crossrefs

Equals 3*A055048(n).

Intersection of A145204 and A189715.

Complement of A055041 with respect to A145204\{0}.

Complement of A055047 with respect to A189715.

Cf. A007913.

Programs

Haskell

a055040 n = a055040_list !! (n-1)
a055040_list = map (* 3) a055048_list
-- Reinhard Zumkeller, Apr 07 2012

Mathematica

max = 500; Select[ Union[ Flatten[ Table[3^(2*i + 1)*(3*j + 2), {i, 0, Ceiling[ Log[max/6]/Log[9]]}, {j, 0, Ceiling[(max/9^i - 6)/9]}]]], # <= max &] (* Jean-François Alcover, Oct 13 2011 *)

Python

from sympy import integer_log
def A055040(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//9**i-2)//3+1 for i in range(integer_log(x,9)[0]+1))
    return bisection(f,n,n)*3 # Chai Wah Wu, Feb 14 2025

Formula

G.f.: [x(x+2)(x^2+x+1)(x^7+x^3+1)]/(x^11-x^10-x+1) (conjectured).