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

0, 1, 4, 13, 67, 229, 229, 958, 958, 7519, 27202, 27202, 204349, 1267231, 1267231, 10833169, 39530983, 125624425, 125624425, 125624425, 1287885892, 4774670293, 15235023496, 46616083105, 140759261932, 140759261932, 988047871375, 3529913699704, 11155511184691

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 A309901(n) modulo 3^n.

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309901.

The 3-adic expansion of exp(3) is given by A317675.

Approximations of exp(p) in p-adic field: this sequence (p=3), A309902 (p=5), A309904 (p=7).

Programs

PARI

a(n) = lift(exp(3 + O(3^n)))