cp's OEIS Frontend

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

%I A055039 #48 Mar 30 2025 03:11:33
%S A055039 14,30,46,56,62,78,94,110,120,126,142,158,174,184,190,206,222,224,238,
%T A055039 248,254,270,286,302,312,318,334,350,366,376,382,398,414,430,440,446,
%U A055039 462,478,480,494,504,510,526,542,558,568,574,590,606,622
%N A055039 Numbers of the form 2^(2i+1)*(8j+7).
%C A055039 The numbers not of the form x^2+y^2+2z^2.
%C A055039 Numbers of the form 6*x^2 + 8*x^2*(2*y -1). (Steve Waterman).
%C A055039 These are the numbers not occurring as norms in the face-centered cubic lattice (cf. A004015).
%C A055039 Numbers whose base 4 representation ends in 3,2 followed by some number of zeros. - _Franklin T. Adams-Watters_, Dec 04 2006
%C A055039 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
%C A055039 The asymptotic density of this sequence is 1/12. - _Amiram Eldar_, Mar 29 2025
%H A055039 T. D. Noe, <a href="/A055039/b055039.txt">Table of n, a(n) for n=1..10000</a>
%H A055039 L. E. Dickson, <a href="http://dx.doi.org/10.1090/S0002-9904-1927-04312-9">Integers represented by positive ternary quadratic forms</a>, Bull. Amer. Math. Soc. 33 (1927), 63-70.
%H A055039 L. J. Mordell, <a href="http://dx.doi.org/10.1093/qmath/os-1.1.276">A new Waring's problem with squares of linear forms</a>, Quart. J. Math., 1 (1930), 276-288 (see p. 283).
%H A055039 Steve Waterman, <a href="http://watermanpolyhedron.com/MISSING.html">Missing numbers formula</a>.
%e A055039 In base 4: 32, 132, 232, 320, 332, 1032, 1132, 1232, 1320, 1332, 2032, ...
%t A055039 Select[Range[650], Mod[# / 4^IntegerExponent[#, 4], 16] == 14 &] (* _Amiram Eldar_, Mar 29 2025 *)
%o A055039 (Python)
%o A055039 from itertools import count, islice
%o A055039 def A055039_gen(startvalue=1): # generator of terms >= startvalue
%o A055039     return filter(lambda n:(m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
%o A055039 A055039_list = list(islice(A055039_gen(),30)) # _Chai Wah Wu_, Jul 09 2022
%o A055039 (Python)
%o A055039 def A055039(n):
%o A055039     def bisection(f,kmin=0,kmax=1):
%o A055039         while f(kmax) > kmax: kmax <<= 1
%o A055039         kmin = kmax >> 1
%o A055039         while kmax-kmin > 1:
%o A055039             kmid = kmax+kmin>>1
%o A055039             if f(kmid) <= kmid:
%o A055039                 kmax = kmid
%o A055039             else:
%o A055039                 kmin = kmid
%o A055039         return kmax
%o A055039     def f(x): return n+x-sum(((x>>i)-7>>3)+1 for i in range(1,x.bit_length(),2))
%o A055039     return bisection(f,n,n) # _Chai Wah Wu_, Feb 24 2025
%Y A055039 Equals twice A004215. Not the same as A044075 - see A124169.
%Y A055039 Complement of A000401.
%Y A055039 Cf. A004015.
%K A055039 nonn,easy
%O A055039 1,1
%A A055039 _N. J. A. Sloane_, Jun 01 2000