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%I #28 Nov 18 2018 10:01:41
%S 0,1,1,5,13,13,13,77,77,333,333,333,333,333,333,16717,16717,16717,
%T 147789,409933,934221,934221,934221,934221,9322829,9322829,9322829,
%U 9322829,143540557,411976013,948846925,948846925,948846925,948846925,9538781517,9538781517
%N Approximation of the 2-adic integer exp(4) up to 2^n.
%C In p-adic field, the exponential function exp(x) is defined as Sum_{k>=0} x^k/k!.
%C 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.
%C When extended to a function over the metric completion of the p-adic field, exp(x) has radius of convergence p^(-1/(p-1)).
%C a(n) is the multiplicative inverse of A321689(n) modulo 2^n. - _Jianing Song_, Nov 17 2018
%H Jianing Song, <a href="/A320814/b320814.txt">Table of n, a(n) for n = 0..2000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%F 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.
%F a(n) = Sum_{i=0..n-1} A320815(i)*2^i.
%e A320840(1) = 1, 4^0/0! = 1, so a(1) = 1.
%e A320840(4) = 3, Sum_{i=0..2} 4^i/i! = 13, so a(4) = 13.
%e A320840(6) = 5, Sum_{i=0..4} 4^i/i! = 103/3 == 13 (mod 64), so a(6) = 13.
%e A320840(8) = 6, Sum_{i=0..5} 4^i/i! = 643/15 == 77 (mod 256), so a(8) = 77.
%e A320840(9) = 7, Sum_{i=0..6} 4^i/i! = 437/9 == 333 (mod 512), so a(9) = 333.
%o (PARI) a(n) = lift(sum(i=0, n-1-(n>=2), Mod(4^i/i!, 2^n)))
%o (PARI) a(n) = lift(exp(4 + O(2^n))); \\ _Andrew Howroyd_, Nov 05 2018
%Y Cf. A000120, A007814, A092391, A320815, A320840, A321689.
%K nonn
%O 0,4
%A _Jianing Song_, Oct 21 2018
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