A319282 Numbers of the form 16^i*(16*j + 15).

15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, 207, 223, 239, 240, 255, 271, 287, 303, 319, 335, 351, 367, 383, 399, 415, 431, 447, 463, 479, 495, 496, 511, 527, 543, 559, 575, 591, 607, 623, 639, 655, 671, 687, 703, 719, 735, 751, 752, 767, 783, 799, 815

Offset: 1

Author

Jianing Song, Sep 16 2018

Keywords

• nonn

Comments

{-a(n)} gives all negative fourth powers modulo all powers of 2, that is, negative fourth powers over 2-adic integers.

Links

• Jianing Song, Table of n, a(n) for n = 1..9999 (all terms <= 150000)

Crossrefs

A125169 is a proper subsequence.

Perfect powers over 2-adic integers:

Squares: positive: A234000; negative: A004215 (negated);

Cubes: A191257;

Fourth powers: positive: A319281; negative: this sequence (negated).

Programs

PARI

isA319282(n)= n\16^valuation(n, 16)%16==15

Python

def A319282(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>>(i<<2))-15)>>4)+1 for i in range(x.bit_length()>>2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 17 2025

Formula

a(n) = 15*n + O(log(n)).