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Expansion of the 2-adic integer Sum_{k>=0} k!.
5

%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