A055043 Numbers of the form 2^(2i+1)*(8*j+3).

6, 22, 24, 38, 54, 70, 86, 88, 96, 102, 118, 134, 150, 152, 166, 182, 198, 214, 216, 230, 246, 262, 278, 280, 294, 310, 326, 342, 344, 352, 358, 374, 384, 390, 406, 408, 422, 438, 454, 470, 472, 486, 502, 518, 534, 536, 550, 566, 582, 598

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. A055046.

Programs

Mathematica

f[upto_]:=Module[{maxi=Floor[(Log[2,upto]-1)/2],maxj= Floor[(upto-3)/8],s},s=2^(2First[#]+1) (8Last[#]+3)&/@ Tuples[{Range[0,maxi], Range[0,maxj]}];Union[Select[s,#<=upto&]]]; f[700]  (* Harvey P. Dale, Mar 23 2011 *)

Python

def A055043(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)-3>>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*A055046(n). - Chai Wah Wu, Mar 19 2025