A341684 Expansion of the 2-adic integer Sum_{k>=0} k!.

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

Jianing Song, Table of n, a(n) for n = 0..1000

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

Adjacent sequences: A341681 A341682 A341683 * A341685 A341686 A341687

Keyword

nonn,base

Author

Jianing Song, Feb 17 2021

Status

approved