A341687 Expansion of the 7-adic integer Sum_{k>=0} k!.

6, 6, 1, 1, 6, 1, 0, 2, 0, 3, 5, 1, 4, 1, 3, 6, 2, 0, 2, 4, 3, 5, 6, 3, 4, 3, 5, 0, 0, 4, 4, 0, 1, 0, 1, 6, 2, 0, 0, 3, 3, 5, 1, 4, 6, 1, 5, 1, 5, 4, 5, 5, 1, 5, 1, 6, 5, 6, 2, 2, 0, 2, 0, 5, 5, 0, 5, 5, 6, 5, 1, 4, 2, 2, 2, 1, 2, 2, 0, 5, 5, 5, 2, 6, 2, 0, 4, 0, 1, 3, 5

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 septenary (base-7) 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 = 1/7^k.

Examples

			Sum_{k>=0} k! = ...33002610104400534365342026314153020161166.
		

Links

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

Crossrefs

Cf. A341683 (successive approximations of Sum_{k>=0} k!).

Expansion of Sum_{k>=0} k! in p-adic integers: A341684 (p=2), A341685 (p=3), A341686 (p=5), this sequence (p=7).

Programs

PARI

a(n) = my(p=7); lift(sum(k=0, (p-1)*((n+1)+logint((p-1)*(n+1), p)), Mod(k!, p^(n+1)))) \ p^n

Formula

a(n) = (A341683(n+1) - A341683(n))/7^n.