A055039 Numbers of the form 2^(2i+1)*(8j+7).

14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 238, 248, 254, 270, 286, 302, 312, 318, 334, 350, 366, 376, 382, 398, 414, 430, 440, 446, 462, 478, 480, 494, 504, 510, 526, 542, 558, 568, 574, 590, 606, 622

Offset: 1

Author

N. J. A. Sloane, Jun 01 2000

Keywords

• nonn
• easy

Comments

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

Numbers of the form 6*x^2 + 8*x^2*(2*y -1). (Steve Waterman).

These are the numbers not occurring as norms in the face-centered cubic lattice (cf. A004015).

Numbers whose base 4 representation ends in 3,2 followed by some number of zeros. - Franklin T. Adams-Watters, Dec 04 2006

Numbers k such that the k-th coefficient of eta(x)^4/eta(x^4) is 0 where eta is the Dedekind eta function. - Benoit Cloitre, Mar 15 2025

The asymptotic density of this sequence is 1/12. - Amiram Eldar, Mar 29 2025

Examples

			In base 4: 32, 132, 232, 320, 332, 1032, 1132, 1232, 1320, 1332, 2032, ...
		

Links

• T. D. Noe, 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.
• L. J. Mordell, A new Waring's problem with squares of linear forms, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
• Steve Waterman, Missing numbers formula.

Crossrefs

Equals twice A004215. Not the same as A044075 - see A124169.

Complement of A000401.

Cf. A004015.

Programs

Mathematica

Select[Range[650], Mod[# / 4^IntegerExponent[#, 4], 16] == 14 &] (* Amiram Eldar, Mar 29 2025 *)

Python

from itertools import count, islice
def A055039_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A055039_list = list(islice(A055039_gen(),30)) # Chai Wah Wu, Jul 09 2022

Python

def A055039(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)-7>>3)+1 for i in range(1,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025