A309975 -id:A309975 - cp's OEIS frontend

A309903 Approximation of the 5-adic integer exp(-5) up to 5^n.

0, 1, 21, 71, 196, 2071, 2071, 33321, 345821, 736446, 8548946, 18314571, 18314571, 994877071, 994877071, 25408939571, 86444095821, 239031986446, 1001971439571, 16260760502071, 92554705814571, 283289569095821, 1236963885502071, 8389521258548946

Offset: 0

Jianing Song, Aug 21 2019

nonn

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 A309902(n) modulo 5^n.

Wikipedia, p-adic number

Cf. A309902.
The 5-adic expansion of exp(5) is given by A309975.
Approximations of exp(-p) in p-adic field: A309901 (p=3), this sequence (p=5), A309905 (p=7).

PARI
```
a(n) = lift(exp(-5 + O(5^n)))
```

A309988 Digits of the 7-adic integer exp(-7).

Original entry on oeis.org

1, 6, 3, 4, 2, 3, 5, 3, 2, 0, 1, 4, 1, 1, 0, 5, 3, 2, 3, 4, 0, 0, 5, 6, 1, 1, 1, 0, 6, 1, 2, 2, 2, 5, 2, 4, 2, 4, 3, 1, 6, 2, 0, 6, 6, 4, 2, 2, 5, 3, 2, 5, 0, 3, 0, 6, 1, 1, 2, 2, 1, 1, 0, 1, 3, 4, 2, 6, 4, 3, 1, 4, 1, 4, 3, 1, 2, 6, 0, 4, 3, 5, 4, 4, 1, 4, 2, 1

Offset: 0

Jianing Song, Aug 26 2019

Digits of the multiplicative inverse of A309987.

Wikipedia, p-adic number

Cf. A309905.

Digits of exp(-p) in p-adic field: A309866 (p=3), A309975 (p=5), this sequence (p=7).

PARI
```
a(n) = lift(exp(-7+O(7^(n+1))))\7^n
```

cp's OEIS Frontend

A309903 Approximation of the 5-adic integer exp(-5) up to 5^n.

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