A321081 -id:A321081 - cp's OEIS frontend

A321080 Approximations up to 2^n for 2-adic integer log_5(-3).

0, 1, 3, 3, 3, 3, 35, 35, 163, 163, 675, 1699, 1699, 1699, 9891, 9891, 42659, 42659, 42659, 304803, 304803, 304803, 304803, 4499107, 4499107, 21276323, 21276323, 21276323, 155494051, 423929507, 423929507, 1497671331, 1497671331, 1497671331, 10087605923

Offset: 2

Jianing Song, Oct 27 2018

nonn

a(n) is the unique number x in [0, 2^(n-2) - 1] such that 5^x == -3 (mod 2^n). This is well defined because {5^x mod 2^n : 0 <= x <= 2^(n-2) - 1} = {1, 5, 9, ..., 2^n - 3}.
For any odd 2-adic integer x, define log(x) = -Sum_{k>=1} (1 - x)^k/k (which always converges over the 2-adic field) and log_x(y) = log(y)/log(x), then we have log(-1) = 0. If we further define exp(x) = Sum_{k>=0} x^k/k! for 2-adic integers divisible by 4, then we have exp(log(x)) = x if and only if x == 1 (mod 4). As a result, log_5(-3) = log_5(3) = log_(-5)(3) = log_(-5)(-3), but it's better to be stated as log_5(-3).
For n > 0, a(n) is also the unique number x in [0, 2^(n-2) - 1] such that (-5)^x == 3 (mod 2^n).
a(n) is the multiplicative inverse of A321082(n) modulo 2^(n-2).

			The only number in the range [0, 2^(n-2) - 1] for n = 2 is 0, so a(2) = 0.
5^a(2) + 3 = 4 which is not divisible by 8, so a(3) = a(2) + 2^0 = 1.
5^a(3) + 3 = 8 which is not divisible by 16, so a(4) = a(3) + 2^1 = 3.
5^a(4) + 3 = 128 which is divisible by 32, 64 and 128 but not 256, so a(5) = a(6) = a(7) = a(4) = 3, a(8) = a(7) + 2^5 = 35.
5^a(8) + 3 = ... which is divisible by 512 but not 1024, so a(9) = a(8) = 35, a(10) = a(9) + 2^7 = 163.

Jianing Song, Table of n, a(n) for n = 2..1000
Wikipedia, p-adic number

Cf. A321081, A321082, A321690, A321691.

b(n) = {my(v=vector(n)); for(n=3, n, v[n] = v[n-1] + if(Mod(5,2^n)^v[n-1] + 3==0, 0, 2^(n-3))); v}
a(n) = b(n)[n] \\ Program provided by Andrew Howroyd

a(n)={if(n<3, 0, truncate(log(-3 + O(2^n))/log(5 + O(2^n))))} \\ Andrew Howroyd, Nov 03 2018

a(2) = 0; for n >= 3, a(n) = a(n-1) if 5^a(n-1) + 3 is divisible by 2^n, otherwise a(n-1) + 2^(n-3).
a(n) = Sum_{i=0..n-3} A321081(i)*2^i (empty sum yields 0 for n = 2).
a(n) = A321691(n+2)/A321690(n+2) mod 2^n.

A321083 Digits of the 2-adic integer log_(-3)(5).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 0

Jianing Song, Oct 27 2018

See A321082 for the definition of log_5(-3) and more information.

Multiplicative inverse of A321081.

			log_(-3)(5) = ...0110111111010101110010000011110100001011.

Jianing Song, Table of n, a(n) for n = 0..1000
Wikipedia, p-adic number

Cf. A321081, A321082, A321694.

b(n) = {my(v=vector(n)); v[3]=1; for(n=4, n, v[n] = v[n-1] + if(Mod(-3,2^n)^v[n-1] - 5==0, 0, 2^(n-3))); v}
a(n) = b(n+3)[n+3]\2^n

a(n) = 0 if (-3)^A321082(n+2) - 5 is divisible by 2^(n+3), otherwise 1.

Equals to A152228/A321694.

cp's OEIS Frontend

A321080 Approximations up to 2^n for 2-adic integer log_5(-3).

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A321083 Digits of the 2-adic integer log_(-3)(5).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula