A055047 Numbers of the form 9^i*(3*j+1).

1, 4, 7, 9, 10, 13, 16, 19, 22, 25, 28, 31, 34, 36, 37, 40, 43, 46, 49, 52, 55, 58, 61, 63, 64, 67, 70, 73, 76, 79, 81, 82, 85, 88, 90, 91, 94, 97, 100, 103, 106, 109, 112, 115, 117, 118, 121, 124, 127, 130, 133, 136, 139, 142, 144, 145, 148, 151, 154, 157, 160, 163

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn

Comments

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

Also values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 3 raised to an odd power. - V. Raman, Dec 18 2013

Numbers whose squarefree part is congruent to 1 modulo 3. - Peter Munn, May 17 2020

Links

• Paolo Xausa, 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 A189715.

Complement of A055048 with respect to A007417.

Complement of A055040 with respect to A189715.

Cf. A007913, A055041, A055046, A233998, A233999.

Programs

Mathematica

A055047Q[k_] := Mod[k/9^IntegerExponent[k, 9], 3] == 1;
Select[Range[300], A055047Q] (* Paolo Xausa, Mar 20 2025 *)

PARI

is(n)=n/=9^valuation(n,9); n%3==1 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

Python

from sympy import integer_log
def A055047(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) # Chai Wah Wu, Feb 14 2025

Formula

a(n) = 8n/3 + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055041(n)/3. - Peter Munn, May 17 2020