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A341687 Expansion of the 7-adic integer Sum_{k>=0} k!. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

LINKS

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

FORMULA

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

EXAMPLE

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

PROG

(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

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).

Sequence in context: A046620 A046619 A021606 * A046606 A172350 A205457

Adjacent sequences:  A341684 A341685 A341686 * A341688 A341689 A341690

KEYWORD

nonn,base

AUTHOR

Jianing Song, Feb 17 2021

STATUS

approved

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Last modified September 16 23:53 EDT 2021. Contains 347477 sequences. (Running on oeis4.)