A309902 Approximation of the 5-adic integer exp(5) up to 5^n.

0, 1, 6, 81, 456, 2956, 6081, 37331, 349831, 1521706, 3474831, 3474831, 101131081, 833552956, 4495662331, 16702693581, 16702693581, 169290584206, 1695169490456, 16953958552956, 55100931209206, 436570657771706, 2343919290584206, 9496476663631081

Offset: 0

Author

Jianing Song, Aug 21 2019

Keywords

• nonn

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 A309903(n) modulo 5^n.

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309903.

The 5-adic expansion of exp(5) is given by A309888.

Approximations of exp(p) in p-adic field: A309900 (p=3), this sequence (p=5), A309904 (p=7).

Programs

PARI

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