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

0, 1, 7, 25, 52, 52, 538, 1267, 1267, 1267, 20950, 20950, 198097, 1260979, 1260979, 6043948, 6043948, 92137390, 92137390, 866978368, 2029239835, 5516024236, 26436730642, 57817790251, 246104147905, 810963220867, 1658251830310, 6741983486968, 21993178456942

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 A309900(n) modulo 3^n.

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309900.

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

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

Programs

PARI

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