A341685 Expansion of the 3-adic integer Sum_{k>=0} k!.

1, 0, 1, 2, 1, 0, 1, 1, 0, 2, 0, 2, 1, 2, 0, 0, 0, 2, 2, 0, 2, 0, 0, 2, 1, 2, 0, 0, 1, 2, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 2, 0, 1, 0, 1, 0, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 1, 1, 0, 0, 1, 2, 2, 1, 1, 1, 0, 2, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1

Offset: 0

Author

Jianing Song, Feb 17 2021

Keywords

• nonn
• base

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 ternary (base-3) 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 = 1/3^k.

Examples

			Sum_{k>=0} k! = ...00210201202210021200202200021202011012101.
		

Links

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

Crossrefs

Cf. A341681 (successive approximations of Sum_{k>=0} k!).

Expansion of Sum_{k>=0} k! in p-adic integers: A341684 (p=2), this sequence (p=3), A341686 (p=5), A341687 (p=7).

Programs

PARI

a(n) = my(p=3); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n

Formula

a(n) = (A341681(n+1) - A341681(n))/3^n.