A321083 -id:A321083 - cp's OEIS frontend

A321081 Digits of the 2-adic integer log_5(-3).

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

Offset: 0

Jianing Song, Oct 27 2018

See A321080 for the definition of log_5(-3) and more information.

Multiplicative inverse of A321083.

			log_5(-3) = ...1000011001011001010001001010011010100011.

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

Cf. A321080, A321083, A321694.

b(n) = {my(v=vector(n)); v[2]=0; 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+3)[n+3]\2^n

a(n) = 0 if 5^A321080(n+2) + 3 is divisible by 2^(n+3), otherwise 1.

Equals to A321694/A152228.

A321082 Approximations up to 2^n for 2-adic integer log_(-3)(5).

Original entry on oeis.org

0, 1, 3, 3, 11, 11, 11, 11, 11, 267, 267, 1291, 3339, 7435, 15627, 15627, 15627, 15627, 15627, 15627, 539915, 539915, 539915, 4734219, 13122827, 29900043, 29900043, 97008907, 97008907, 365444363, 365444363, 1439186187, 3586669835, 7881637131, 16471571723

Offset: 2

Jianing Song, Oct 27 2018

nonn

a(n) is the unique number x in [0, 2^(n-2) - 1] such that (-3)^x == 5 (mod 2^n). This is well defined because {(-3)^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_(-3)(5) = log_(-3)(-5) = log_3(5) = log_3(-5), but it's better to be stated as log_(-3)(5).
For n > 0, a(n) is also the unique number x in [0, 2^(n-2) - 1] such that 3^x == -5 (mod 2^n).
a(n) is the multiplicative inverse of A321080(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.
(-3)^a(2) - 5 = -4 which is not divisible by 8, so a(3) = a(2) + 2^0 = 1.
(-3)^a(3) - 5 = -8 which is not divisible by 16, so a(4) = a(3) + 2^1 = 3.
(-3)^a(4) - 5 = -32 which is divisible by 32 but not 64, so a(5) = a(4) = 3, a(6) = a(5) + 2^3 = 11.
(-3)^a(6) - 5 = -177152 which is divisible by 128, 256, 512, 1024 but not 2048, so a(7) = a(8) = a(9) = a(10) = a(6) = 11, a(11) = a(10) + 2^8 = 267.

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

Cf. A321080, A321083, A321690, A321691.

b(n) = {my(v=vector(n)); v[2]=0; for(n=3, 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)[n]

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

a(2) = 0; for n >= 3, a(n) = a(n-1) if (-3)^a(n-1) - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-3).
a(n) = Sum_{i=0..n-3} A321083(i)*2^i (empty sum yields 0 for n = 2).
a(n) = A321690(n+2)/A321691(n+2) mod 2^n.

cp's OEIS Frontend

A321081 Digits of the 2-adic integer log_5(-3).

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A321082 Approximations up to 2^n for 2-adic integer log_(-3)(5).

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