A341686 Expansion of the 5-adic integer Sum_{k>=0} k!.

4, 2, 2, 2, 4, 0, 0, 3, 2, 0, 1, 0, 3, 1, 3, 4, 0, 0, 2, 3, 4, 3, 1, 1, 0, 2, 1, 3, 4, 0, 3, 1, 0, 2, 4, 0, 1, 2, 1, 1, 1, 1, 0, 1, 3, 2, 0, 1, 1, 1, 4, 2, 3, 2, 0, 3, 3, 0, 0, 3, 0, 1, 4, 2, 0, 0, 2, 4, 0, 4, 3, 3, 1, 3, 3, 1, 4, 4, 2, 4, 2, 3, 4, 1, 0, 4, 0, 2, 4, 2, 2

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 quinary (base-5) 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 = 1/5^k.

Examples

			Sum_{k>=0} k! = ...11121042013043120113432004313010230042224.
		

Links

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

Crossrefs

Cf. A341682 (successive approximations of Sum_{k>=0} k!).

Expansion of Sum_{k>=0} k! in p-adic integers: A341684 (p=2), A341685 (p=3), this sequence (p=5), A341687 (p=7).

Programs

PARI

a(n) = my(p=5); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n

Formula

a(n) = (A341682(n+1) - A341682(n))/5^n.