A319293 Numbers of the form 27^i*(9*j +- 1).

1, 8, 10, 17, 19, 26, 27, 28, 35, 37, 44, 46, 53, 55, 62, 64, 71, 73, 80, 82, 89, 91, 98, 100, 107, 109, 116, 118, 125, 127, 134, 136, 143, 145, 152, 154, 161, 163, 170, 172, 179, 181, 188, 190, 197, 199, 206, 208, 215, 216, 217, 224, 226, 233, 235, 242, 244

Offset: 1

Author

Jianing Song, Sep 16 2018

Keywords

• nonn

Comments

{+-a(n)} gives all nonzero cubes modulo all powers of 3, that is, cubes over 3-adic integers. So this sequence is closed under multiplication.

Links

• Jianing Song, Table of n, a(n) for n = 1..9231 (all terms <= 40000)

Crossrefs

A056020 is a proper subsequence.

Perfect powers over 3-adic integers:

Squares: positive: A055047; negative: A055048 (negated);

Cubes: this sequence.

Programs

PARI

isA319293(n)= n\27^valuation(n, 27)%9==1||n\27^valuation(n, 27)%9==8

Python

from sympy import integer_log
def A319293(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(((m:=x//27**i)-1)//9+(m-8)//9+2 for i in range(integer_log(x,27)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

Formula

a(n) = 13*n/3 + O(log(n)).