cp's OEIS Frontend

A055042 Numbers of the form 2^(2i+1)*(8*j+5).

%I A055042 #28 Apr 21 2025 23:05:21
%S A055042 10,26,40,42,58,74,90,104,106,122,138,154,160,168,170,186,202,218,232,
%T A055042 234,250,266,282,296,298,314,330,346,360,362,378,394,410,416,424,426,
%U A055042 442,458,474,488,490,506,522,538,552,554,570,586,602,616
%N A055042 Numbers of the form 2^(2i+1)*(8*j+5).
%C A055042 These are also the numbers not of the form x^2+2y^2+3z^2.
%C A055042 The asymptotic density of this sequence is 1/12. - _Amiram Eldar_, Mar 29 2025
%D A055042 Burton W. Jones, The Arithmetic of Quadratic Forms, Carus Monograph 10, Math. Assoc. America, 1967; Problem 60, p. 204.
%H A055042 David A. Corneth, <a href="/A055042/b055042.txt">Table of n, a(n) for n = 1..10000</a>
%H A055042 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).
%F A055042 a(n) = 2*A055045(n). - _Chai Wah Wu_, Mar 19 2025
%e A055042 42 = 21*2 = 2^(2*0 + 1)*(8*2 + 5) is in the sequence. - _David A. Corneth_, Apr 18 2021
%t A055042 With[{max = 700}, Flatten[Table[2^(2*i + 1)*(8*j + 5), {i, 0, (Log2[max] - 1)/2}, {j, 0, Floor[(max/2^(2*i + 1) - 5)/8]}]] // Sort] (* _Amiram Eldar_, Mar 29 2025 *)
%o A055042 (PARI) upto(n) = { my(res = List()); for(i = 0, logint(n\2, 2), forstep(j = 5, n>>(2*i+1), 8, listput(res, 4^i*2*j) ) ); Set(res) } \\ _David A. Corneth_, Apr 18 2021
%o A055042 (Python)
%o A055042 def A055042(n):
%o A055042     def bisection(f,kmin=0,kmax=1):
%o A055042         while f(kmax) > kmax: kmax <<= 1
%o A055042         kmin = kmax >> 1
%o A055042         while kmax-kmin > 1:
%o A055042             kmid = kmax+kmin>>1
%o A055042             if f(kmid) <= kmid:
%o A055042                 kmax = kmid
%o A055042             else:
%o A055042                 kmin = kmid
%o A055042         return kmax
%o A055042     def f(x): return n+x-sum(((x>>(i<<1)+1)-5>>3)+1 for i in range(x.bit_length()-1>>1))
%o A055042     return bisection(f,n,n) # _Chai Wah Wu_, Mar 19 2025
%Y A055042 Cf. A055045.
%K A055042 nonn,easy
%O A055042 1,1
%A A055042 _N. J. A. Sloane_, Jun 01 2000