A055044 Numbers of the form 2^(2i+1)*(8*j+1).

2, 8, 18, 32, 34, 50, 66, 72, 82, 98, 114, 128, 130, 136, 146, 162, 178, 194, 200, 210, 226, 242, 258, 264, 274, 288, 290, 306, 322, 328, 338, 354, 370, 386, 392, 402, 418, 434, 450, 456, 466, 482, 498, 512, 514, 520, 530, 544, 546, 562, 578

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn
• easy

Comments

The asymptotic density of this sequence is 1/12. - Amiram Eldar, Mar 29 2025

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

Cf. A234000.

Programs

Mathematica

With[{max = 600}, Flatten[Table[2^(2*i + 1)*(8*j + 1), {i, 0, (Log2[max] - 1)/2}, {j, 0, Floor[(max/2^(2*i + 1) - 1)/8]}]] // Sort] (* Amiram Eldar, Mar 29 2025 *)

Python

def A055044(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<<1)+1)-1>>3)+1 for i in range(x.bit_length()+1>>1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 19 2025

Formula

a(n) = 2*A234000(n). - Chai Wah Wu, Mar 19 2025