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A321690
Approximations up to 2^n for the 2-adic integer log(5).
4
0, 0, 0, 4, 12, 28, 60, 124, 124, 124, 636, 1660, 1660, 1660, 9852, 9852, 9852, 9852, 140924, 140924, 140924, 1189500, 3286652, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 7480956, 2154964604, 2154964604, 2154964604, 19334833788
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>=1} (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).
FORMULA
a(n) = Sum_{i=0..n-1} A152228(i)*2^i.
EXAMPLE
a(3) = (4 + O(2^3)) mod 8 = 4 mod 8 = 4.
a(6) = (4 - 4^2/2 + O(2^6)) mod 64 = (-4) mod 64 = 60.
a(10) = (4 - 4^2/2 + 4^3/3 - 4^4/4 + O(2^10)) mod 1024 = (-140/3) mod 1024 = 636.
a(11) = (4 - 4^2/2 + 4^3/3 - 4^4/4 + 4^5/5 + O(2^11)) mod 2048 = (2372/15) mod 2048 = 1660.
PROG
(PARI) a(n) = if(n, lift(log(5 + O(2^n))), 0);
CROSSREFS
Cf. A321691 (log(-3)).
Sequence in context: A186924 A261320 A179023 * A269712 A028399 A173033
KEYWORD
nonn
AUTHOR
Jianing Song, Nov 17 2018
STATUS
approved