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

0, 1, 21, 71, 196, 2071, 2071, 33321, 345821, 736446, 8548946, 18314571, 18314571, 994877071, 994877071, 25408939571, 86444095821, 239031986446, 1001971439571, 16260760502071, 92554705814571, 283289569095821, 1236963885502071, 8389521258548946

Offset

0,3

Comments

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.

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

Links

Table of n, a(n) for n=0..23.

Wikipedia, p-adic number

Prog

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

Crossrefs

Cf. A309902.

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

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

Sequence in context: A195026 A296035 A102233 * A187719 A156285 A160435

Adjacent sequences: A309900 A309901 A309902 * A309904 A309905 A309906

Keyword

nonn

Author

Jianing Song, Aug 21 2019

Status

approved