cp's OEIS Frontend

A055045 Numbers of the form 4^i*(8*j+5).

%I A055045 #55 Mar 20 2025 08:53:10
%S A055045 5,13,20,21,29,37,45,52,53,61,69,77,80,84,85,93,101,109,116,117,125,
%T A055045 133,141,148,149,157,165,173,180,181,189,197,205,208,212,213,221,229,
%U A055045 237,244,245,253,261,269,276,277,285,293,301,308,309,317,320,325,333,336,340,341
%N A055045 Numbers of the form 4^i*(8*j+5).
%C A055045 Numbers not of the form x^2+2y^2+6z^2.
%H A055045 Reinhard Zumkeller, <a href="/A055045/b055045.txt">Table of n, a(n) for n = 1..10000</a>
%H A055045 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. See Theorem XI.
%H A055045 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 A055045 a(n) = 6n + O(log n). - _Charles R Greathouse IV_, Dec 19 2013
%F A055045 a(n) = A055042(n)/2. - _Chai Wah Wu_, Mar 19 2025
%t A055045 A055045Q[k_] := Mod[k/4^IntegerExponent[k, 4], 8] == 5;
%t A055045 Select[Range[500], A055045Q] (* _Paolo Xausa_, Mar 20 2025 *)
%o A055045 (PARI) is(n)=n/=4^valuation(n,4); n%8==5 \\ _Charles R Greathouse IV_ and _V. Raman_, Dec 19 2013
%o A055045 (Haskell)
%o A055045 a055045 n = a055045_list !! (n-1)
%o A055045 a055045_list = filter ((== 5) . (flip mod 8) . f) [1..] where
%o A055045    f x = if r == 0 then f x' else x  where (x', r) = divMod x 4
%o A055045 -- _Reinhard Zumkeller_, Jan 02 2014
%o A055045 (Python)
%o A055045 from itertools import count, islice
%o A055045 def A055045_gen(startvalue=1): # generator of terms >= startvalue
%o A055045     return filter(lambda n:not (m:=(~n&n-1).bit_length())&1 and (n>>m)&7==5, count(max(startvalue, 1)))
%o A055045 A055045_list = list(islice(A055045_gen(), 30)) # _Chai Wah Wu_, Jul 09 2022
%o A055045 (Python)
%o A055045 def A055045(n):
%o A055045     def bisection(f,kmin=0,kmax=1):
%o A055045         while f(kmax) > kmax: kmax <<= 1
%o A055045         kmin = kmax >> 1
%o A055045         while kmax-kmin > 1:
%o A055045             kmid = kmax+kmin>>1
%o A055045             if f(kmid) <= kmid:
%o A055045                 kmax = kmid
%o A055045             else:
%o A055045                 kmin = kmid
%o A055045         return kmax
%o A055045     def f(x): return n+x-sum(((x>>(i<<1))-5>>3)+1 for i in range(x.bit_length()>>1))
%o A055045     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A055045 Cf. A004215, A055042, A055046, A234000.
%K A055045 nonn,easy
%O A055045 1,1
%A A055045 _N. J. A. Sloane_, Jun 01 2000