A309866 -id:A309866 - cp's OEIS frontend

A317675 Coefficients of 3-adic expansion of exp(3).

1, 1, 1, 2, 2, 0, 1, 0, 1, 1, 0, 1, 2, 0, 2, 2, 2, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 2, 0, 1, 2, 0, 1, 1, 0, 0, 2, 2, 1, 1, 2, 1, 0, 1, 2, 0, 1, 1, 1, 2, 0, 2, 2, 2, 0, 1, 0, 2, 0, 0, 1, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 2, 0, 0, 0, 1

Offset: 0

Robert Israel, Aug 03 2018

			exp(3) = 1 + 3 + 9/2 + 9/2 + 27/8 + ... = 1 + 3 + 3^2 + 2*3^3 + O(3^4).

Robert Israel, Table of n, a(n) for n = 0..10000
Wikipedia, p-adic exponential

Cf. A091933, A309866.

Maple
```
R:= evalp(exp(3),3,200):
op([1,3],R);
```

N=100; Vecrev(digits(lift(exp(3+O(3^N))), 3), N) \\ Seiichi Manyama, Aug 20 2019

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

Original entry on oeis.org

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

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 A309900(n) modulo 3^n.

Wikipedia, p-adic number

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).

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

A309975 Digits of the 5-adic integer exp(-5).

Original entry on oeis.org

1, 4, 2, 1, 3, 0, 2, 4, 1, 4, 1, 0, 4, 0, 4, 2, 1, 1, 4, 4, 2, 2, 3, 2, 2, 4, 3, 4, 4, 2, 0, 0, 1, 0, 4, 3, 2, 1, 3, 2, 0, 4, 3, 2, 4, 4, 1, 4, 0, 0, 4, 3, 4, 3, 0, 4, 3, 4, 1, 2, 4, 1, 3, 3, 3, 4, 3, 2, 4, 4, 3, 2, 4, 4, 3, 2, 4, 3, 4, 2, 2, 2, 0, 2, 3, 1, 3, 2

Offset: 0

Jianing Song, Aug 26 2019

Digits of the multiplicative inverse of A309888.

Wikipedia, p-adic number

Cf. A309903.

Digits of exp(-p) in p-adic field: A309866 (p=3), this sequence (p=5), A309988 (p=7).

PARI
```
a(n) = lift(exp(-5+O(5^(n+1))))\5^n
```

cp's OEIS Frontend

A317675 Coefficients of 3-adic expansion of exp(3).

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

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

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A309975 Digits of the 5-adic integer exp(-5).

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