A055050 Numbers of the form 4^i*(8*j+3) or 4^i*(8*j+7).

3, 7, 11, 12, 15, 19, 23, 27, 28, 31, 35, 39, 43, 44, 47, 48, 51, 55, 59, 60, 63, 67, 71, 75, 76, 79, 83, 87, 91, 92, 95, 99, 103, 107, 108, 111, 112, 115, 119, 123, 124, 127, 131, 135, 139, 140, 143, 147, 151, 155, 156, 159, 163, 167, 171, 172

Offset: 1

Author

N. J. A. Sloane, Jun 02 2000

Keywords

• nonn

Comments

The numbers not of the form x^2+y^2+4z^2.

Positions of 3 in A065882. - Clark Kimberling, Oct 19 2016

The asymptotic density of this sequence is 1/3. - Amiram Eldar, Mar 08 2021

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• M. A. Bennett and B. Reznick, Positive rational solutions to x^y = y^{mx}: : A Number-Theoretic Excursion, Amer. Math. Monthly, 111 (No. 1, 2004), 13-21.
• L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).

Programs

Mathematica

t = Table[Mod[n/4^IntegerExponent[n, 4], 4], {n, 1, 160}] (*A065882*)
Flatten[Position[t, 3]] (*A055050*) (* Clark Kimberling, Oct 19 2016 *)

Python

def A055050(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>>(i<<1))-3>>3)+(m-7>>3)+2 for i in range((x.bit_length()>>1)+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 21 2025