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A341684
Expansion of the 2-adic integer Sum_{k>=0} k!.
5
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, 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, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1
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
FORMULA
a(n) = (A341680(n+1) - A341680(n))/2^n.
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
Cf. A341680 (successive approximations of Sum_{k>=0} k!).
Expansion of Sum_{k>=0} k! in p-adic integers: this sequence (p=2), A341685 (p=3), A341686 (p=5), A341687 (p=7).
Sequence in context: A115516 A285830 A288741 * A327183 A347870 A188967
KEYWORD
nonn,base
AUTHOR
Jianing Song, Feb 17 2021
STATUS
approved