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

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

0,4

Comments

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.

Links

Table of n, a(n) for n=0..34.

Wikipedia, p-adic number

Formula

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.

Example

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.

Prog

(PARI) 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)));

Crossrefs

Cf. A320814, A320840, A321692.

Sequence in context: A077307 A181943 A243757 * A369340 A368870 A076568

Adjacent sequences: A321686 A321687 A321688 * A321690 A321691 A321692

Keyword

nonn

Author

Jianing Song, Nov 17 2018

Status

approved