A055041 Numbers of the form 3^(2i+1)*(3*j+1).

3, 12, 21, 27, 30, 39, 48, 57, 66, 75, 84, 93, 102, 108, 111, 120, 129, 138, 147, 156, 165, 174, 183, 189, 192, 201, 210, 219, 228, 237, 243, 246, 255, 264, 270, 273, 282, 291, 300, 309, 318, 327, 336, 345, 351, 354, 363, 372, 381, 390, 399

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn

Comments

The numbers not of the form x^2+y^2+6z^2.

Numbers whose squarefree part is congruent to 3 modulo 9. Compare with A329575. - Peter Munn, May 17 2020

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

Links

• Amiram Eldar, 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 A145204 and A189716.

Complement of A055040 with respect to A145204\{0}.

Complement of A055048 with respect to A189716.

Cf. A007913, A055047, A329575.

Programs

Mathematica

f[p_, e_] := (p^Mod[e, 2]); sqfpart[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[400], Mod[sqfpart[#], 9] == 3 &] (* Amiram Eldar, Mar 08 2021 *)

Python

from sympy import integer_log
def A055041(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-1)//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

a(n) = A055047(n) * 3. - Peter Munn, May 17 2020