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

1,1

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).

Formula

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

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 *)

Prog

(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
(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

Crossrefs

Intersection of A007417 and A189716.

Complement of A055047 with respect to A007417.

Complement of A055041 with respect to A189716.

Cf. A007913, A055040.

Sequence in context: A356057 A189525 A189369 * A045930 A140099 A276385

Adjacent sequences: A055045 A055046 A055047 * A055049 A055050 A055051

Keyword

nonn

Author

N. J. A. Sloane, Jun 01 2000

Status

approved