A309905 Approximation of the 7-adic integer exp(-7) up to 7^n.

0, 1, 43, 190, 1562, 6364, 56785, 645030, 3115659, 14645261, 14645261, 297120510, 8206427482, 22047714683, 118936725090, 118936725090, 23856744274805, 123555535983608, 588816563958022, 5474057357689369, 51069638099181941, 51069638099181941

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 A309904(n) modulo 7^n.

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A309904.

The 7-adic expansion of exp(-7) is given by A309988.

Approximations of exp(-p) in p-adic field: A309901 (p=3), A309903 (p=5), this sequence (p=7).

Programs

PARI

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