A320840 -id:A320840 - cp's OEIS frontend

A320814 Approximation of the 2-adic integer exp(4) up to 2^n.

0, 1, 1, 5, 13, 13, 13, 77, 77, 333, 333, 333, 333, 333, 333, 16717, 16717, 16717, 147789, 409933, 934221, 934221, 934221, 934221, 9322829, 9322829, 9322829, 9322829, 143540557, 411976013, 948846925, 948846925, 948846925, 948846925, 9538781517, 9538781517

Offset: 0

Jianing Song, Oct 21 2018

nonn

In p-adic field, the exponential function exp(x) is defined as Sum_{k>=0} x^k/k!.
Let |x|A007814(x)%20be%20the%202-adic%20valuation%20of%20x.%20For%20any%202-adic%20number%20x,%20exp(x)%20=%20Sum">2 be the 2-adic metric of x, and v(x, 2) = A007814(x) be the 2-adic valuation of x. For any 2-adic number x, exp(x) = Sum{i>=0} x^i/i! converges implies that lim_{k->+oo} |x^k/k!|2 = 0, that is, lim{k->+oo} v(x^k/k!, 2) = +oo, or lim_{k->+oo} (k*(v(x, 2) - 1) + A000120(i)) = +oo. So v(x, 2) > 1, x is a 2-adic integer divisible by 4. On the other hand, for any integer n and i >= A320840(n), v(4^i/i!, 2) = A092391(i) >= n, so approximation of exp(4) up to 2^n is wholly determined by Sum_{i=0..A320840(n)-1} 4^i/i! (see formula section below), which is well-defined because it has only finitely many terms.
When extended to a function over the metric completion of the p-adic field, exp(x) has radius of convergence p^(-1/(p-1)).
a(n) is the multiplicative inverse of A321689(n) modulo 2^n. - Jianing Song, Nov 17 2018

			A320840(1) = 1, 4^0/0! = 1, so a(1) = 1.
A320840(4) = 3, Sum_{i=0..2} 4^i/i! = 13, so a(4) = 13.
A320840(6) = 5, Sum_{i=0..4} 4^i/i! = 103/3 == 13 (mod 64), so a(6) = 13.
A320840(8) = 6, Sum_{i=0..5} 4^i/i! = 643/15 == 77 (mod 256), so a(8) = 77.
A320840(9) = 7, Sum_{i=0..6} 4^i/i! = 437/9 == 333 (mod 512), so a(9) = 333.

Jianing Song, Table of n, a(n) for n = 0..2000
Wikipedia, p-adic number

Cf. A000120, A007814, A092391, A320815, A320840, A321689.

a(n) = lift(sum(i=0, n-1-(n>=2), Mod(4^i/i!, 2^n)))

a(n) = lift(exp(4 + O(2^n))); \\ Andrew Howroyd, Nov 05 2018

If Sum_{i=0..A320840(n)-1} 4^i/i! = p/q, gcd(p, q) = 1, then a(n) = p*q^(-1) mod 2^n.

a(n) = Sum_{i=0..n-1} A320815(i)*2^i.

A321689 Approximation of the 2-adic integer exp(-4) up to 2^n.

Original entry on oeis.org

0, 1, 1, 5, 5, 5, 5, 5, 133, 389, 901, 1925, 3973, 8069, 8069, 24453, 57221, 57221, 188293, 450437, 974725, 974725, 974725, 974725, 974725, 17751941, 17751941, 84860805, 84860805, 84860805, 621731717, 621731717, 621731717, 4916699013, 4916699013

Offset: 0

Jianing Song, Nov 17 2018

nonn

Let 4Q_2 = {x belongs to Q_2 : |x|2 <= 1/4} and 4Q_2 + 1 = {x belongs to Q_2: |x - 1|_2 <= 1/4}. Define exp(x) = Sum{k>=0} x^k/k! and log(x) = -Sum_{k>=0} (1 - x)^k/k over 2-adic field, then exp(x) is a one-to-one mapping from 4Q_2 to 4Q_2 + 1, and log(x) is the inverse of exp(x).

a(n) is the multiplicative inverse of A320814(n) modulo 2^n.

			A320840(1) = 1, (-4)^0/0! = 1, so a(1) = 1.
A320840(3) = 2, Sum_{i=0..1} (-4)^i/i! = -3 == 5 (mod 8), so a(3) = 5.
A320840(8) = 6, Sum_{i=0..5} (-4)^i/i! = -53/15 == 133 (mod 256), so a(8) = 133.
A320840(9) = 7, Sum_{i=0..6} (-4)^i/i! = 97/45 == 389 (mod 512), so a(9) = 389.
A320840(10) = 9, Sum_{i=0..8} (-4)^i/i! = 167/315 == 901 (mod 1024), so a(10) = 901.

Wikipedia, p-adic number

Cf. A320814, A320840, A321692.

a(n) = lift(sum(i=0, n-1-(n>=2), Mod((-4)^i/i!, 2^n)))

PARI
```
a(n) = lift(exp(-4 + O(2^n)));
```

If Sum_{i=0..A320840(n)-1} (-4)^i/i! = p/q, gcd(p, q) = 1, then a(n) = p*q^(-1) mod 2^n.

a(n) = Sum_{i=0..n-1} A321692(i)*2^i.

cp's OEIS Frontend

A320814 Approximation of the 2-adic integer exp(4) up to 2^n.

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A321689 Approximation of the 2-adic integer exp(-4) up to 2^n.

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Author

Keywords

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Examples

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PARI

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Formula