%I #11 Feb 19 2021 03:38:41
%S 0,1,0,1,1,0,0,0,0,1,0,1,1,1,1,1,1,0,1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,
%T 1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,1,1,1,
%U 0,1,0,0,0,0,0,0,1,1,0,0,1,1,1,0,0,0,0,0,0,0,1
%N Expansion of the 2-adic integer Sum_{k>=0} k!.
%C 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 Conjecture: this constant is transcendental, which means that it is not the root of any polynomial with integer coefficients.
%C Conjecture: this constant is normal, which means for every binary (base-2) 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 = 1/2^k.
%H Jianing Song, <a href="/A341684/b341684.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (A341680(n+1) - A341680(n))/2^n.
%e Sum_{k>=0} k! = ...10010110111111000011111011111101000011010.
%o (PARI) a(n) = my(p=2); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n
%Y Cf. A341680 (successive approximations of Sum_{k>=0} k!).
%Y Expansion of Sum_{k>=0} k! in p-adic integers: this sequence (p=2), A341685 (p=3), A341686 (p=5), A341687 (p=7).
%K nonn,base
%O 0
%A _Jianing Song_, Feb 17 2021