cp's OEIS Frontend

A055048 Numbers of the form 9^i*(3*j+2).

%I A055048 #46 Feb 14 2025 17:30:18
%S A055048 2,5,8,11,14,17,18,20,23,26,29,32,35,38,41,44,45,47,50,53,56,59,62,65,
%T A055048 68,71,72,74,77,80,83,86,89,92,95,98,99,101,104,107,110,113,116,119,
%U A055048 122,125,126,128,131,134,137,140,143,146,149,152,153,155
%N A055048 Numbers of the form 9^i*(3*j+2).
%C A055048 The numbers not of the form x^2+3y^2+3z^2.
%C A055048 Numbers whose squarefree part is congruent to 2 modulo 3. - _Peter Munn_, May 17 2020
%C A055048 The asymptotic density of this sequence is 3/8. - _Amiram Eldar_, Mar 08 2021
%H A055048 Reinhard Zumkeller, <a href="/A055048/b055048.txt">Table of n, a(n) for n = 1..10000</a>
%H A055048 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 A055048 a(n) = A055040(n)/3. - _Peter Munn_, May 17 2020
%t A055048 max = 200; Select[ Union[ Flatten[ Table[ 9^i*(3*j + 2), {i, 0, Ceiling[Log[max]/Log[9]]}, {j, 0, Ceiling[( max/9^i - 2)/3]}]]], # <= max &] (* _Jean-François Alcover_, Oct 13 2011 *)
%o A055048 (Haskell)
%o A055048 a055048 n = a055048_list !! (n-1)
%o A055048 a055048_list = filter (s 0) [1..] where
%o A055048    s t u | m > 0  = even t && m == 2
%o A055048          | m == 0 = s (t + 1) u' where (u',m) = divMod u 3
%o A055048 -- _Reinhard Zumkeller_, Apr 07 2012
%o A055048 (PARI) is(n)=n/=9^valuation(n, 9); n%3==2 \\ _Charles R Greathouse IV_ and _V. Raman_, Dec 19 2013
%o A055048 (Python)
%o A055048 from sympy import integer_log
%o A055048 def A055048(n):
%o A055048     def bisection(f,kmin=0,kmax=1):
%o A055048         while f(kmax) > kmax: kmax <<= 1
%o A055048         kmin = kmax >> 1
%o A055048         while kmax-kmin > 1:
%o A055048             kmid = kmax+kmin>>1
%o A055048             if f(kmid) <= kmid:
%o A055048                 kmax = kmid
%o A055048             else:
%o A055048                 kmin = kmid
%o A055048         return kmax
%o A055048     def f(x): return n+x-sum((x//9**i-2)//3+1 for i in range(integer_log(x,9)[0]+1))
%o A055048     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A055048 Intersection of A007417 and A189716.
%Y A055048 Complement of A055047 with respect to A007417.
%Y A055048 Complement of A055041 with respect to A189716.
%Y A055048 Cf. A007913, A055040.
%K A055048 nonn
%O A055048 1,1
%A A055048 _N. J. A. Sloane_, Jun 01 2000