OFFSET
0
COMMENTS
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.
Conjecture: this constant is transcendental, which means that it is not the root of any polynomial with integer coefficients.
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.
LINKS
Jianing Song, Table of n, a(n) for n = 0..1000
EXAMPLE
Sum_{k>=0} k! = ...10010110111111000011111011111101000011010.
PROG
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Jianing Song, Feb 17 2021
STATUS
approved