A309904 Approximation of the 7-adic integer exp(7) up to 7^n.

0, 1, 8, 204, 890, 890, 51311, 286609, 3580781, 20875184, 182289612, 747240110, 8656547082, 8656547082, 105545557489, 783768630338, 15026453160167, 114725244868970, 1045247300817798, 9187315290370043, 20586210475743186, 20586210475743186

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 A309905(n) modulo 7^n.

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309905.

The 7-adic expansion of exp(7) is given by A309987.

Approximations of exp(p) in p-adic field: A309900 (p=3), A309902 (p=5), this sequence (p=7).

Programs

PARI

a(n) = lift(exp(7 + O(7^n)))