A055045 Numbers of the form 4^i*(8*j+5).

5, 13, 20, 21, 29, 37, 45, 52, 53, 61, 69, 77, 80, 84, 85, 93, 101, 109, 116, 117, 125, 133, 141, 148, 149, 157, 165, 173, 180, 181, 189, 197, 205, 208, 212, 213, 221, 229, 237, 244, 245, 253, 261, 269, 276, 277, 285, 293, 301, 308, 309, 317, 320, 325, 333, 336, 340, 341

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn
• easy

Comments

Numbers not of the form x^2+2y^2+6z^2.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. See Theorem XI.
• 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, A055042, A055046, A234000.

Programs

Haskell

a055045 n = a055045_list !! (n-1)
a055045_list = filter ((== 5) . (flip mod 8) . f) [1..] where
   f x = if r == 0 then f x' else x  where (x', r) = divMod x 4
-- Reinhard Zumkeller, Jan 02 2014

Mathematica

A055045Q[k_] := Mod[k/4^IntegerExponent[k, 4], 8] == 5;
Select[Range[500], A055045Q] (* Paolo Xausa, Mar 20 2025 *)

PARI

is(n)=n/=4^valuation(n,4); n%8==5 \\ Charles R Greathouse IV and V. Raman, Dec 19 2013

Python

from itertools import count, islice
def A055045_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==5, count(max(startvalue, 1)))
A055045_list = list(islice(A055045_gen(), 30)) # Chai Wah Wu, Jul 09 2022

Python

def A055045(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))-5>>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) = A055042(n)/2. - Chai Wah Wu, Mar 19 2025