cp's OEIS Frontend

A055047 Numbers of the form 9^i*(3*j+1).

%I A055047 #57 Mar 20 2025 08:53:20
%S A055047 1,4,7,9,10,13,16,19,22,25,28,31,34,36,37,40,43,46,49,52,55,58,61,63,
%T A055047 64,67,70,73,76,79,81,82,85,88,90,91,94,97,100,103,106,109,112,115,
%U A055047 117,118,121,124,127,130,133,136,139,142,144,145,148,151,154,157,160,163
%N A055047 Numbers of the form 9^i*(3*j+1).
%C A055047 The numbers not of the form 2x^2+3y^2+3z^2.
%C A055047 Also values of n such that numbers of the form x^2+n*y^2 for some integers x, y cannot have prime factor of 3 raised to an odd power. - _V. Raman_, Dec 18 2013
%C A055047 Numbers whose squarefree part is congruent to 1 modulo 3. - _Peter Munn_, May 17 2020
%H A055047 Paolo Xausa, <a href="/A055047/b055047.txt">Table of n, a(n) for n = 1..10000</a>
%H A055047 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 A055047 a(n) = 8n/3 + O(log n). - _Charles R Greathouse IV_, Dec 19 2013
%F A055047 a(n) = A055041(n)/3. - _Peter Munn_, May 17 2020
%t A055047 A055047Q[k_] := Mod[k/9^IntegerExponent[k, 9], 3] == 1;
%t A055047 Select[Range[300], A055047Q] (* _Paolo Xausa_, Mar 20 2025 *)
%o A055047 (PARI) is(n)=n/=9^valuation(n,9); n%3==1 \\ _Charles R Greathouse IV_ and _V. Raman_, Dec 19 2013
%o A055047 (Python)
%o A055047 from sympy import integer_log
%o A055047 def A055047(n):
%o A055047     def bisection(f,kmin=0,kmax=1):
%o A055047         while f(kmax) > kmax: kmax <<= 1
%o A055047         kmin = kmax >> 1
%o A055047         while kmax-kmin > 1:
%o A055047             kmid = kmax+kmin>>1
%o A055047             if f(kmid) <= kmid:
%o A055047                 kmax = kmid
%o A055047             else:
%o A055047                 kmin = kmid
%o A055047         return kmax
%o A055047     def f(x): return n+x-sum((x//9**i-1)//3+1 for i in range(integer_log(x,9)[0]+1))
%o A055047     return bisection(f,n,n) # _Chai Wah Wu_, Feb 14 2025
%Y A055047 Intersection of A007417 and A189715.
%Y A055047 Complement of A055048 with respect to A007417.
%Y A055047 Complement of A055040 with respect to A189715.
%Y A055047 Cf. A007913, A055041, A055046, A233998, A233999.
%K A055047 nonn
%O A055047 1,2
%A A055047 _N. J. A. Sloane_, Jun 01 2000