A055052 Numbers of the form 4^i*(8j+7) or 4^i*(8j+5).

5, 7, 13, 15, 20, 21, 23, 28, 29, 31, 37, 39, 45, 47, 52, 53, 55, 60, 61, 63, 69, 71, 77, 79, 80, 84, 85, 87, 92, 93, 95, 101, 103, 109, 111, 112, 116, 117, 119, 124, 125, 127, 133, 135, 141, 143, 148, 149, 151, 156, 157, 159, 165, 167, 173, 175

Offset: 1

Author

N. J. A. Sloane, Jun 02 2000

Keywords

• nonn
• easy

Comments

Numbers not of the form x^2+2y^2+8z^2.

The integers that are ratios between the terms constitute the sequence's complement within A003159. - Peter Munn, Feb 07 2024

The asymptotic density of this sequence is 1/3. - Amiram Eldar, Feb 09 2024

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

Disjoint union of A004215 and A055045.

Subsequence of A003159, A097700.

Programs

Mathematica

Select[Range[200], MemberQ[{5, 7}, Mod[# / 4^IntegerExponent[#, 4], 8]] &] (* Amiram Eldar, Feb 09 2024 *)

Python

def A055052(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):
        c = n+x
        for i in range(x.bit_length()>>1):
            m = x>>(i<<1)
            c -= (m-5>>3)+(m-7>>3)+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025