A055048 Numbers of the form 9^i*(3*j+2).

2, 5, 8, 11, 14, 17, 18, 20, 23, 26, 29, 32, 35, 38, 41, 44, 45, 47, 50, 53, 56, 59, 62, 65, 68, 71, 72, 74, 77, 80, 83, 86, 89, 92, 95, 98, 99, 101, 104, 107, 110, 113, 116, 119, 122, 125, 126, 128, 131, 134, 137, 140, 143, 146, 149, 152, 153, 155

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn

Comments

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

Numbers whose squarefree part is congruent to 2 modulo 3. - Peter Munn, May 17 2020

The asymptotic density of this sequence is 3/8. - 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

Intersection of A007417 and A189716.

Complement of A055047 with respect to A007417.

Complement of A055041 with respect to A189716.

Cf. A007913, A055040.

Programs

Haskell

a055048 n = a055048_list !! (n-1)
a055048_list = filter (s 0) [1..] where
   s t u | m > 0  = even t && m == 2
         | m == 0 = s (t + 1) u' where (u',m) = divMod u 3
-- Reinhard Zumkeller, Apr 07 2012

Mathematica

max = 200; Select[ Union[ Flatten[ Table[ 9^i*(3*j + 2), {i, 0, Ceiling[Log[max]/Log[9]]}, {j, 0, Ceiling[( max/9^i - 2)/3]}]]], # <= max &] (* Jean-François Alcover, Oct 13 2011 *)

PARI

is(n)=n/=9^valuation(n, 9); n%3==2 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

Python

from sympy import integer_log
def A055048(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) # Chai Wah Wu, Feb 14 2025

Formula

a(n) = A055040(n)/3. - Peter Munn, May 17 2020