A320814 - OEIS

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

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OFFSET

0,4

COMMENTS

In p-adic field, the exponential function exp(x) is defined as Sum_{k>=0} x^k/k!.

Let |x|_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

LINKS

Jianing Song, Table of n, a(n) for n = 0..2000

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} A320815(i)*2^i.

EXAMPLE

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.