cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A321690 Approximations up to 2^n for the 2-adic integer log(5).

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 17 2018

Keywords

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).

Examples

			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.

Links

Crossrefs

Cf. A321691 (log(-3)).

Programs

  • PARI
    a(n) = if(n, lift(log(5 + O(2^n))), 0);

Formula

a(n) = Sum_{i=0..n-1} A152228(i)*2^i.

A321694 Digits of the 2-adic integer log(-3).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1
Offset: 0

Views

Author

Jianing Song, Nov 17 2018

Keywords

Comments

Appears to equal A309754 shifted right one place. - R. J. Mathar, Jul 27 2023

Examples

			log(-3) = ...01110001100100010100100101110100011110100.

Links

Crossrefs

Cf. A321691, A152228 (log(5)).

Programs

  • PARI
    a(n) = lift(log(-3 + O(2^(n+1))))\2^n;

Formula

a(n) = (A321691(n+1) - A321691(n))/2^n.

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

Views

Author

Jianing Song, Oct 27 2018

Keywords

Comments

See A321080 for the definition of log_5(-3) and more information.
Multiplicative inverse of A321083.

Examples

			log_5(-3) = ...1000011001011001010001001010011010100011.

Links

Crossrefs

Cf. A321080, A321083, A321694.

Programs

  • PARI
    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

Formula

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

Views

Author

Jianing Song, Oct 27 2018

Keywords

Comments

See A321082 for the definition of log_5(-3) and more information.
Multiplicative inverse of A321081.

Examples

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

Links

Crossrefs

Cf. A321081, A321082, A321694.

Programs

  • PARI
    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

Formula

a(n) = 0 if (-3)^A321082(n+2) - 5 is divisible by 2^(n+3), otherwise 1.
Equals to A152228/A321694.
Showing 1-4 of 4 results.

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