cp's OEIS Frontend

A273893 Denominator of n/3^n.

%I A273893 #31 Mar 13 2020 17:44:29
%S A273893 1,3,9,9,81,243,243,2187,6561,2187,59049,177147,177147,1594323,
%T A273893 4782969,4782969,43046721,129140163,43046721,1162261467,3486784401,
%U A273893 3486784401,31381059609,94143178827,94143178827,847288609443,2541865828329,282429536481,22876792454961
%N A273893 Denominator of n/3^n.
%C A273893 The reduced values are Ms(n) = 0, 1/3, 2/9, 1/9, 4/81, 5/243, 2/243, 7/2187, 8/6561, 1/2187, ... .
%C A273893 Numerators: 0, 1, 2, 1, 4, ... = A038502(n).
%C A273893 Ms(-n) = 0, -3, -18, ... = - A036290(n).
%C A273893 Difference table of Ms(n):
%C A273893 0,          1/3,    2/9,    1/9,   4/81, 5/243, 2/243, ...
%C A273893 1/3,       -1/9,   -1/9,  -5/81, -7/243, -1/81, ...
%C A273893 -4/9,         0,   4/81,  8/243,  4/243, ...
%C A273893 4/9,       4/81, -4/243, -4/243, ...
%C A273893 -32/81, -16/243,      0, ...
%C A273893 80/243,  16/243, ...
%C A273893 -64/243, ...
%C A273893 etc.
%C A273893 The difference table of O(n) = n/2^n (Oresme numbers) has its 0's on the main diagonal. Here the 0's appear every two rows. For n/4^n,they appear every three rows. (The denominators of O(n) are 2^A093048(n)).
%C A273893 All terms are powers of 3 (A000244).
%F A273893 For n>0, a(n) = 3^(n - valuation(n,3)) = 3^(n - A007949(n)). - _Tom Edgar_, Jun 02 2016
%F A273893 a(3n+1) = 3^(3n+1), a(3n+2) = 3^(3n+2).
%F A273893 a(3n+6) = 27*(3n+3).
%F A273893 From _Peter Bala_, Feb 25 2019: (Start)
%F A273893 a(n) = 3^n/gcd(n,3^n).
%F A273893 O.g.f.: 1 + F(3*x) - (2/3)*F((3*x)^3) - (2/9)*F((3*x)^9) - (2/27)*F((3*x)^27) - ..., where F(x) = x/(1 - x).
%F A273893 O.g.f. for reciprocals: Sum_{n >= 0} x^n/a(n) = 1 +  F((x/3)) + 2*( F((x/3)^3) + 3*F((x/3)^9) + 9*F((x/3)^27) + ... ). Cf. A038502. (End)
%t A273893 Table[Denominator[n/3^n], {n, 0, 28}] (* _Michael De Vlieger_, Jun 03 2016 *)
%o A273893 (Sage) [1] + [3^(n-n.valuation(3)) for n in [1..30]] # _Tom Edgar_, Jun 02 2016
%o A273893 (PARI) a(n) = denominator(n/3^n) \\ _Felix Fröhlich_, Jun 07 2016
%Y A273893 Cf. A007949, A000244, A013732, A013733, A036290, A038502, A075101.
%K A273893 nonn,frac
%O A273893 0,2
%A A273893 _Paul Curtz_, Jun 02 2016