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A309903 Approximation of the 5-adic integer exp(-5) up to 5^n. 4


%S 0,1,21,71,196,2071,2071,33321,345821,736446,8548946,18314571,

%T 18314571,994877071,994877071,25408939571,86444095821,239031986446,

%U 1001971439571,16260760502071,92554705814571,283289569095821,1236963885502071,8389521258548946

%N Approximation of the 5-adic integer exp(-5) up to 5^n.

%C In p-adic field, the exponential function exp(x) is defined as Sum_{k>=0} x^k/k!. When extended to a function over the metric completion of the p-adic field, exp(x) has radius of convergence p^(-1/(p-1)) (i.e., exp(x) converges for x such that |x|_p < p^(-1/(p-1)), where |x|_p is the p-adic metric). As a result, for odd primes p, exp(p) is well-defined in p-adic field, and exp(4) is well defined in 2-adic field.

%C a(n) is the multiplicative inverse of A309902(n) modulo 5^n.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>

%o (PARI) a(n) = lift(exp(-5 + O(5^n)))

%Y Cf. A309902.

%Y The 5-adic expansion of exp(5) is given by A309975.

%Y Approximations of exp(-p) in p-adic field: A309901 (p=3), this sequence (p=5), A309905 (p=7).

%K nonn

%O 0,3

%A _Jianing Song_, Aug 21 2019

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)