cp's OEIS Frontend

A055052 Numbers of the form 4^i*(8j+7) or 4^i*(8j+5).

%I A055052 #27 Feb 14 2025 17:25:36
%S A055052 5,7,13,15,20,21,23,28,29,31,37,39,45,47,52,53,55,60,61,63,69,71,77,
%T A055052 79,80,84,85,87,92,93,95,101,103,109,111,112,116,117,119,124,125,127,
%U A055052 133,135,141,143,148,149,151,156,157,159,165,167,173,175
%N A055052 Numbers of the form 4^i*(8j+7) or 4^i*(8j+5).
%C A055052 Numbers not of the form x^2+2y^2+8z^2.
%C A055052 The integers that are ratios between the terms constitute the sequence's complement within A003159. - _Peter Munn_, Feb 07 2024
%C A055052 The asymptotic density of this sequence is 1/3. - _Amiram Eldar_, Feb 09 2024
%H A055052 Amiram Eldar, <a href="/A055052/b055052.txt">Table of n, a(n) for n = 1..10000</a>
%H A055052 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).
%t A055052 Select[Range[200], MemberQ[{5, 7}, Mod[# / 4^IntegerExponent[#, 4], 8]] &] (* _Amiram Eldar_, Feb 09 2024 *)
%o A055052 (Python)
%o A055052 def A055052(n):
%o A055052     def bisection(f,kmin=0,kmax=1):
%o A055052         while f(kmax) > kmax: kmax <<= 1
%o A055052         kmin = kmax >> 1
%o A055052         while kmax-kmin > 1:
%o A055052             kmid = kmax+kmin>>1
%o A055052             if f(kmid) <= kmid:
%o A055052                 kmax = kmid
%o A055052             else:
%o A055052                 kmin = kmid
%o A055052         return kmax
%o A055052     def f(x):
%o A055052         c = n+x
%o A055052         for i in range(x.bit_length()>>1):
%o A055052             m = x>>(i<<1)
%o A055052             c -= (m-5>>3)+(m-7>>3)+2
%o A055052         return c
%o A055052     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A055052 Disjoint union of A004215 and A055045.
%Y A055052 Subsequence of A003159, A097700.
%K A055052 nonn,easy
%O A055052 1,1
%A A055052 _N. J. A. Sloane_, Jun 02 2000