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 A341686 #11 Feb 19 2021 03:38:17
%S A341686 4,2,2,2,4,0,0,3,2,0,1,0,3,1,3,4,0,0,2,3,4,3,1,1,0,2,1,3,4,0,3,1,0,2,
%T A341686 4,0,1,2,1,1,1,1,0,1,3,2,0,1,1,1,4,2,3,2,0,3,3,0,0,3,0,1,4,2,0,0,2,4,
%U A341686 0,4,3,3,1,3,3,1,4,4,2,4,2,3,4,1,0,4,0,2,4,2,2
%N A341686 Expansion of the 5-adic integer Sum_{k>=0} k!.
%C A341686 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 A341686 Conjecture: this constant is transcendental, which means that it is not the root of any polynomial with integer coefficients.
%C A341686 Conjecture: this constant is normal, which means for every quinary (base-5) 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 = 1/5^k.
%H A341686 Jianing Song, <a href="/A341686/b341686.txt">Table of n, a(n) for n = 0..1000</a>
%F A341686 a(n) = (A341682(n+1) - A341682(n))/5^n.
%e A341686 Sum_{k>=0} k! = ...11121042013043120113432004313010230042224.
%o A341686 (PARI) a(n) = my(p=5); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n
%Y A341686 Cf. A341682 (successive approximations of Sum_{k>=0} k!).
%Y A341686 Expansion of Sum_{k>=0} k! in p-adic integers: A341684 (p=2), A341685 (p=3), this sequence (p=5), A341687 (p=7).
%K A341686 nonn,base
%O A341686 0,1
%A A341686 _Jianing Song_, Feb 17 2021