A321694 -id:A321694 - cp's OEIS frontend

A152228 2-adic expansion of log(5).

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

Offset: 0

Paul D. Hanna, Nov 29 2008

			log(5) (2-adic) = ...0001110110010010000000011100100010011001111100 (base 2).
log(5) (2-adic) = 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^9 + 2^10 + 2^13 +...
exp( log(5) (2-adic) ) = 101 (base 2) = 5.

Wikipedia, p-adic number

Cf. A068434, A321690, A321694 (log(-3)).

a(n)=(truncate( log(5+2*O(2^n)) )%2^(n+1))\2^n

a(n) = (A321690(n+1) - A321690(n))/2^n. - Jianing Song, Nov 17 2018

A321691 Approximations up to 2^n for the 2-adic integer log(-3).

Original entry on oeis.org

0, 0, 0, 4, 4, 20, 52, 116, 244, 244, 244, 244, 2292, 2292, 10484, 26868, 59636, 59636, 190708, 190708, 190708, 1239284, 1239284, 1239284, 9627892, 9627892, 43182324, 43182324, 43182324, 43182324, 580053236, 580053236, 580053236, 4875020532

Offset: 0

Jianing Song, Nov 17 2018

nonn

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).

			a(3) = (-4 + O(2^3)) mod 8 = (-4) mod 8 = 4.
a(6) = (-4 - 4^2/2 - O(2^6)) mod 64 = (-12) mod 64 = 52.
a(10) = (-4 - 4^2/2 - 4^3/3 - 4^4/4 - O(2^10)) mod 1024 = (-292/3) mod 1024 = 244.
a(11) = (-4 - 4^2/2 - 4^3/3 - 4^4/4 - 4^5/5 - O(2^11)) mod 2048 = (-4532/15) mod 2048 = 244.

Wikipedia, p-adic number

Cf. A321690 (log(5)), A321694.

a(n) = if(n, lift(log(-3 + O(2^n))), 0);

a(n) = Sum_{i=0..n-1} A321694(i)*2^i.

Conjecture: a(n) = 2*A309753(n-1). - R. J. Mathar, Aug 06 2023

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

Original entry on oeis.org

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.

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.

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A152228 2-adic expansion of log(5).

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A321691 Approximations up to 2^n for the 2-adic integer log(-3).

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

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

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