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 A341685 #11 Feb 19 2021 03:38:21
%S A341685 1,0,1,2,1,0,1,1,0,2,0,2,1,2,0,0,0,2,2,0,2,0,0,2,1,2,0,0,1,2,2,0,2,1,
%T A341685 0,2,0,1,2,0,0,1,0,0,1,1,1,1,1,0,2,0,2,2,0,1,0,1,0,1,2,1,2,1,2,0,1,2,
%U A341685 1,1,1,0,0,1,2,2,1,1,1,0,2,0,1,0,0,2,0,0,2,2,1
%N A341685 Expansion of the 3-adic integer Sum_{k>=0} k!.
%C A341685 For every prime p, since valuation(k!,p) goes to infinity as k increases, Sum_{k>=0} k! is a well-defined p-adic constant.
%C A341685 Conjecture: this constant is transcendental, which means that it is not the root of any polynomial with integer coefficients.
%C A341685 Conjecture: this constant is normal, which means for every ternary (base-3) string s with length k, if we denote N(s,n) as the number of occurrences of s in the first n digits, then lim_{n->inf} N(s,n)/n = 1/3^k.
%H A341685 Jianing Song, <a href="/A341685/b341685.txt">Table of n, a(n) for n = 0..1000</a>
%F A341685 a(n) = (A341681(n+1) - A341681(n))/3^n.
%e A341685 Sum_{k>=0} k! = ...00210201202210021200202200021202011012101.
%o A341685 (PARI) a(n) = my(p=3); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n
%Y A341685 Cf. A341681 (successive approximations of Sum_{k>=0} k!).
%Y A341685 Expansion of Sum_{k>=0} k! in p-adic integers: A341684 (p=2), this sequence (p=3), A341686 (p=5), A341687 (p=7).
%K A341685 nonn,base
%O A341685 0,4
%A A341685 _Jianing Song_, Feb 17 2021