cp's OEIS Frontend

A055044 Numbers of the form 2^(2i+1)*(8*j+1).

%I A055044 #22 Apr 21 2025 23:05:30
%S A055044 2,8,18,32,34,50,66,72,82,98,114,128,130,136,146,162,178,194,200,210,
%T A055044 226,242,258,264,274,288,290,306,322,328,338,354,370,386,392,402,418,
%U A055044 434,450,456,466,482,498,512,514,520,530,544,546,562,578
%N A055044 Numbers of the form 2^(2i+1)*(8*j+1).
%C A055044 The asymptotic density of this sequence is 1/12. - _Amiram Eldar_, Mar 29 2025
%H A055044 Amiram Eldar, <a href="/A055044/b055044.txt">Table of n, a(n) for n = 1..10000</a>
%H A055044 L. J. Mordell, <a href="https://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).
%F A055044 a(n) = 2*A234000(n). - _Chai Wah Wu_, Mar 19 2025
%t A055044 With[{max = 600}, Flatten[Table[2^(2*i + 1)*(8*j + 1), {i, 0, (Log2[max] - 1)/2}, {j, 0, Floor[(max/2^(2*i + 1) - 1)/8]}]] // Sort] (* _Amiram Eldar_, Mar 29 2025 *)
%o A055044 (Python)
%o A055044 def A055044(n):
%o A055044     def bisection(f,kmin=0,kmax=1):
%o A055044         while f(kmax) > kmax: kmax <<= 1
%o A055044         kmin = kmax >> 1
%o A055044         while kmax-kmin > 1:
%o A055044             kmid = kmax+kmin>>1
%o A055044             if f(kmid) <= kmid:
%o A055044                 kmax = kmid
%o A055044             else:
%o A055044                 kmin = kmid
%o A055044         return kmax
%o A055044     def f(x): return n+x-sum(((x>>(i<<1)+1)-1>>3)+1 for i in range(x.bit_length()+1>>1))
%o A055044     return bisection(f,n,n) # _Chai Wah Wu_, Mar 19 2025
%Y A055044 Cf. A234000.
%K A055044 nonn,easy
%O A055044 1,1
%A A055044 _N. J. A. Sloane_, Jun 01 2000