A055046 Numbers of the form 4^i*(8*j+3).

3, 11, 12, 19, 27, 35, 43, 44, 48, 51, 59, 67, 75, 76, 83, 91, 99, 107, 108, 115, 123, 131, 139, 140, 147, 155, 163, 171, 172, 176, 179, 187, 192, 195, 203, 204, 211, 219, 227, 235, 236, 243, 251, 259, 267, 268, 275, 283, 291, 299, 300, 304, 307, 315, 323, 331, 332

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn
• easy

Comments

Numbers not of the form x^2+y^2+5z^2.

Also values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 2 raised to an odd power. - V. Raman, Dec 18 2013

Links

• Paolo Xausa, 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. A004215, A055043, A055045, A055047, A233998, A233999, A234000.

Programs

Mathematica

A055046Q[k_] := Mod[k/4^IntegerExponent[k, 4], 8] == 3;
Select[Range[500], A055046Q] (* Paolo Xausa, Mar 20 2025 *)

PARI

is(n)=n/=4^valuation(n,4); n%8==3 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

Python

from itertools import count, islice
def A055046_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==3,count(max(startvalue,1)))
A055046_list = list(islice(A055046_gen(),30)) # Chai Wah Wu, Jul 09 2022

Python

def A055046(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))-3>>3)+1 for i in range(x.bit_length()>>1))
    return bisection(f,n,n) # Chai Wah Wu, Feb 14 2025

Formula

a(n) = 6n + O(log n). - Charles R Greathouse IV, Dec 19 2013

a(n) = A055043(n)/2. - Chai Wah Wu, Mar 19 2025