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

1,2

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).

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

Mathematica

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

Prog

(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

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.

Sequence in context: A010389 A010415 A010442 * A234429 A391356 A291201

Adjacent sequences: A055044 A055045 A055046 * A055048 A055049 A055050

Keyword

nonn

Author

N. J. A. Sloane, Jun 01 2000

Status

approved