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A309901 Approximation of the 3-adic integer exp(-3) up to 3^n. 3


%S 0,1,7,25,52,52,538,1267,1267,1267,20950,20950,198097,1260979,1260979,

%T 6043948,6043948,92137390,92137390,866978368,2029239835,5516024236,

%U 26436730642,57817790251,246104147905,810963220867,1658251830310,6741983486968,21993178456942

%N Approximation of the 3-adic integer exp(-3) up to 3^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 A309900(n) modulo 3^n.

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

%o (PARI) a(n) = lift(exp(-3 + O(3^n)))

%Y Cf. A309900.

%Y The 3-adic expansion of exp(-3) is given by A309866.

%Y Approximations of exp(-p) in p-adic field: this sequence (p=3), A309903 (p=5), A309905 (p=7).

%K nonn

%O 0,3

%A _Jianing Song_, Aug 21 2019

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Last modified August 4 12:54 EDT 2021. Contains 346447 sequences. (Running on oeis4.)