This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049803 #47 Dec 16 2019 10:22:47
%S A049803 0,0,0,1,2,0,1,2,0,2,4,3,5,7,6,8,10,0,2,4,3,5,7,6,8,10,0,3,6,6,9,12,
%T A049803 12,15,18,9,12,15,15,18,21,21,24,27,18,21,24,24,27,30,30,33,36,0,3,6,
%U A049803 6,9,12,12,15,18,9,12,15,15,18,21,21,24,27,18
%N A049803 a(n) = n mod 3 + n mod 9 + ... + n mod 3^k, where 3^k <= n < 3^(k+1).
%C A049803 From _Petros Hadjicostas_, Dec 11 2019: (Start)
%C A049803 Conjecture: For b >= 2, consider the function s(n,b) = Sum_{1 <= b^j <= n} (n mod b^j) from p. 8 in Dearden et al. (2011). Then s(b*n + r, b) = b*s(n,b) + r*N(n,b) for 0 <= r <= b-1, where N(n,b) = floor(log_b(n)) + 1 is the number of digits in the base-b representation of n. As initial conditions, we have s(n,b) = 0 for 1 <= n <= b. (This is a generalization of a result by _Robert Israel_ in A049802.)
%C A049803 Here b = 3 and a(n) = s(n,3).
%C A049803 We have N(n,2) = A070939(n), N(n,3) = A081604(n), N(n,4) = A110591(n), and N(n,5) = A110592(n).
%C A049803 If A_b(x) = Sum_{n >= 1} s(n,b)*x^n is the g.f. of the sequence (s(n,b): n >= 1) and the above conjecture is correct, then it can be proved that A_b(x) = b * A_b(x^b) * (1-x^b)/(1-x) + x * ((b-1)*x^b - b*x^(b-1) + 1)/((1-x)^2 * (1-x^b)) * Sum_{k >= 1} x^(b^k). (End)
%H A049803 Metin Sariyar, <a href="/A049803/b049803.txt">Table of n, a(n) for n = 1..32000</a>
%H A049803 B. Dearden, J. Iiams, and J. Metzger, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Dearden/dearden3r.html">A Function Related to the Rumor Sequence Conjecture</a>, J. Int. Seq. 14 (2011), #11.2.3, Example 7.
%F A049803 From _Petros Hadjicostas_, Dec 11 2019: (Start)
%F A049803 Conjecture: a(3*n+r) = 3*a(n) + r*A081604(n) = 3*a(n) + r*(floor(log_3(n)) + 1) for n >= 1 and r = 0, 1, 2.
%F A049803 If the conjecture above is true, the g.f. A(x) satisfies A(x) = 3*(1+x+x^2)*A(x^3) + x*(2*x+1)/(1-x^3) * Sum_{k >= 1} x^(3^k). (End)
%p A049803 a:= n-> add(irem(n, 3^j), j=1..ilog[3](n)):
%p A049803 seq(a(n), n=1..105); # _Alois P. Heinz_, Dec 13 2019
%t A049803 Table[n * Floor@Log[3, n] - Sum[Floor[n*3^-k]*3^k, {k, Log[3, n]}], {n, 100}] (* after _Federico Provvedi_ in A049802*) (* _Metin Sariyar_, Dec 12 2019 *)
%o A049803 (PARI) a(n) = sum(k=1, logint(n, 3), n % 3^k); \\ _Michel Marcus_, Dec 12 2019
%Y A049803 Cf. A049802, A049804, A070939, A081604, A110591, A110592, A330358.
%K A049803 nonn
%O A049803 1,5
%A A049803 _Clark Kimberling_