cp's OEIS Frontend

A055050 Numbers of the form 4^i*(8*j+3) or 4^i*(8*j+7).

%I A055050 #22 Apr 21 2025 23:05:34
%S A055050 3,7,11,12,15,19,23,27,28,31,35,39,43,44,47,48,51,55,59,60,63,67,71,
%T A055050 75,76,79,83,87,91,92,95,99,103,107,108,111,112,115,119,123,124,127,
%U A055050 131,135,139,140,143,147,151,155,156,159,163,167,171,172
%N A055050 Numbers of the form 4^i*(8*j+3) or 4^i*(8*j+7).
%C A055050 The numbers not of the form x^2+y^2+4z^2.
%C A055050 Positions of 3 in A065882. - _Clark Kimberling_, Oct 19 2016
%C A055050 The asymptotic density of this sequence is 1/3. - _Amiram Eldar_, Mar 08 2021
%H A055050 Clark Kimberling, <a href="/A055050/b055050.txt">Table of n, a(n) for n = 1..10000</a>
%H A055050 M. A. Bennett and B. Reznick, <a href="http://www.jstor.org/stable/4145011">Positive rational solutions to x^y = y^{mx}: : A Number-Theoretic Excursion</a>, Amer. Math. Monthly, 111 (No. 1, 2004), 13-21.
%H A055050 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).
%t A055050 t = Table[Mod[n/4^IntegerExponent[n, 4], 4], {n, 1, 160}] (*A065882*)
%t A055050 Flatten[Position[t, 3]] (*A055050*) (* _Clark Kimberling_, Oct 19 2016 *)
%o A055050 (Python)
%o A055050 def A055050(n):
%o A055050     def bisection(f,kmin=0,kmax=1):
%o A055050         while f(kmax) > kmax: kmax <<= 1
%o A055050         kmin = kmax >> 1
%o A055050         while kmax-kmin > 1:
%o A055050             kmid = kmax+kmin>>1
%o A055050             if f(kmid) <= kmid:
%o A055050                 kmax = kmid
%o A055050             else:
%o A055050                 kmin = kmid
%o A055050         return kmax
%o A055050     def f(x): return n+x-sum(((m:=x>>(i<<1))-3>>3)+(m-7>>3)+2 for i in range((x.bit_length()>>1)+1))
%o A055050     return bisection(f,n,n) # _Chai Wah Wu_, Mar 21 2025
%K A055050 nonn
%O A055050 1,1
%A A055050 _N. J. A. Sloane_, Jun 02 2000