A341685 - OEIS

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

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OFFSET

0,4

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.

LINKS

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

FORMULA

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

EXAMPLE

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

PROG

(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

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).

Sequence in context: A025915 A081285 A255361 * A069844 A233006 A145152

Adjacent sequences: A341682 A341683 A341684 * A341686 A341687 A341688

KEYWORD

nonn,base

AUTHOR

Jianing Song, Feb 17 2021

STATUS

approved