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.

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A324029 Digits of one of the two 5-adic integers sqrt(-6) that is related to A324027.

Original entry on oeis.org

2, 2, 1, 1, 2, 3, 2, 4, 3, 1, 0, 0, 1, 3, 1, 3, 4, 2, 3, 2, 3, 2, 4, 4, 2, 3, 3, 0, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 3, 4, 1, 0, 2, 4, 4, 3, 4, 0, 3, 2, 0, 2, 0, 2, 0, 3, 2, 0, 0, 4, 2, 4, 4, 0, 4, 4, 4, 3, 1, 4, 2, 2, 4, 2, 0, 0, 0, 3, 0, 4, 3, 2, 4, 3, 3, 4, 0
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

This square root of -6 in the 5-adic field ends with digit 2. The other, A324030, ends with digit 3.

Examples

			The solution to x^2 == -6 (mod 5^4) such that x == 2 (mod 5) is x == 162 (mod 5^4), and 162 is written as 1122 in quinary, so the first four terms are 2, 2, 1 and 1.

Links

Crossrefs

Cf. A324027, A324028.
Digits of 5-adic square roots:
this sequence, A324030 (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
A324025, A324026 (sqrt(6)).

Programs

  • PARI
    a(n) = truncate(sqrt(-6+O(5^(n+1))))\5^n

Formula

a(n) = (A324027(n+1) - A324027(n))/5^n.
For n > 0, a(n) = 4 - A324030(n).
Equals A210850*A324026 = A210851*A324025, where each A-number represents a 5-adic number.

A324030 Digits of one of the two 5-adic integers sqrt(-6) that is related to A324028.

Original entry on oeis.org

3, 2, 3, 3, 2, 1, 2, 0, 1, 3, 4, 4, 3, 1, 3, 1, 0, 2, 1, 2, 1, 2, 0, 0, 2, 1, 1, 4, 3, 3, 1, 3, 3, 3, 1, 3, 2, 1, 2, 1, 0, 3, 4, 2, 0, 0, 1, 0, 4, 1, 2, 4, 2, 4, 2, 4, 1, 2, 4, 4, 0, 2, 0, 0, 4, 0, 0, 0, 1, 3, 0, 2, 2, 0, 2, 4, 4, 4, 1, 4, 0, 1, 2, 0, 1, 1, 0, 4
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

This square root of -6 in the 5-adic field ends with digit 3. The other, A324029, ends with digit 2.

Examples

			The solution to x^2 == -6 (mod 5^4) such that x == 3 (mod 5) is x == 463 (mod 5^4), and 463 is written as 3323 in quinary, so the first four terms are 3, 2, 3 and 3.

Links

Crossrefs

Cf. A324027, A324028.
Digits of 5-adic square roots:
A324029, sequence (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
A324025, A324026 (sqrt(6)).

Programs

  • PARI
    a(n) = truncate(-sqrt(-6+O(5^(n+1))))\5^n

Formula

a(n) = (A324028(n+1) - A324028(n))/5^n.
For n > 0, a(n) = 4 - A324029(n).
Equals A210850*A324025 = A210851*A324026, where each A-number represents a 5-adic number.

A327305 Digits of one of the two 5-adic integers sqrt(-9) that is related to A327303.

Original entry on oeis.org

4, 0, 3, 0, 0, 1, 1, 4, 2, 0, 2, 2, 3, 2, 4, 4, 1, 1, 2, 2, 3, 0, 2, 2, 4, 2, 1, 4, 1, 4, 0, 0, 0, 2, 4, 1, 1, 3, 1, 1, 0, 4, 1, 2, 1, 2, 2, 1, 1, 2, 0, 0, 3, 1, 2, 0, 4, 2, 0, 3, 4, 4, 0, 0, 0, 0, 1, 4, 0, 3, 4, 0, 1, 4, 4, 3, 3, 0, 2, 3, 2, 3, 3, 3, 1, 4, 2, 4
Offset: 0

Views

Author

Jianing Song, Sep 16 2019

Keywords

Comments

This is the 5-adic solution to x^2 = -9 that ends in 4. A327304 gives the other solution that ends in 1.

Examples

			Equals ...1131142000414124220322114423220241100304.

Links

Crossrefs

Cf. A327302, A327303.
Digits of 5-adic square roots:
A327304, this sequence (sqrt(-9));
A324029, A324030 (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
A324025, A324026 (sqrt(6)).

Programs

  • Maple
    op([1,1,3], select(t -> padic:-ratvaluep(t,1)=4, [padic:-rootp(x^2+9,5,100)])); # Robert Israel, Aug 31 2020
  • PARI
    a(n) = truncate(-sqrt(-9+O(5^(n+1))))\5^n

Formula

For n > 0, a(n) is the unique m in {0, 1, 2, 3, 4} such that (A327303(n) + m*5^n)^2 + 9 is divisible by 5^(n+1).
a(n) = (A327303(n+1) - A327303(n))/5^n.
For n > 0, a(n) = 4 - A327304(n).

A327304 Digits of one of the two 5-adic integers sqrt(-9) that is related to A327302.

Original entry on oeis.org

1, 4, 1, 4, 4, 3, 3, 0, 2, 4, 2, 2, 1, 2, 0, 0, 3, 3, 2, 2, 1, 4, 2, 2, 0, 2, 3, 0, 3, 0, 4, 4, 4, 2, 0, 3, 3, 1, 3, 3, 4, 0, 3, 2, 3, 2, 2, 3, 3, 2, 4, 4, 1, 3, 2, 4, 0, 2, 4, 1, 0, 0, 4, 4, 4, 4, 3, 0, 4, 1, 0, 4, 3, 0, 0, 1, 1, 4, 2, 1, 2, 1, 1, 1, 3, 0, 2, 0
Offset: 0

Views

Author

Jianing Song, Sep 16 2019

Keywords

Comments

This is the 5-adic solution to x^2 = -9 that ends in 1. A327305 gives the other solution that ends in 4.

Examples

			Equals ...3313302444030320224122330021224203344141.

Links

Crossrefs

Cf. A327302, A327303.
Digits of 5-adic square roots:
this sequence, A327305 (sqrt(-9));
A324029, A324030 (sqrt(-6));
A269591, A269592 (sqrt(-4));
A210850, A210851 (sqrt(-1));
A324025, A324026 (sqrt(6)).

Programs

  • PARI
    a(n) = truncate(-sqrt(-9+O(5^(n+1))))\5^n

Formula

For n > 0, a(n) is the unique m in {0, 1, 2, 3, 4} such that (A327302(n) + m*5^n)^2 + 9 is divisible by 5^(n+1).
a(n) = (A327302(n+1) - A327302(n))/5^n.
For n > 0, a(n) = 4 - A327305(n).

A051276 Nonzero coefficients in one of the 5-adic expansions of sqrt(-1).

Original entry on oeis.org

2, 1, 2, 1, 3, 4, 2, 3, 3, 2, 2, 4, 1, 3, 2, 4, 4, 3, 4, 4, 1, 2, 4, 1, 4, 1, 1, 3, 1, 4, 1, 4, 2, 1, 1, 3, 3, 2, 2, 4, 4, 2, 4, 3, 1, 2, 4, 3, 3, 3, 3, 1, 3, 1, 1, 3, 3, 4, 1, 3, 3, 3, 4, 2, 2, 2, 1, 4, 1, 1, 4, 4, 2, 1, 2, 3, 4, 4, 4, 2, 2, 1, 3, 1, 3, 2, 4, 2, 1, 4, 3, 4, 3, 1, 2, 1, 3, 3, 3, 1, 1, 3, 1, 2, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			2 + 1*5 + 2*5^2 + 1*5^3 + 3*5^4 + 4*5^5 + 2*5^6 + 3*5^7 + 3*5^9 + 2*5^10 + 2*5^11 + 4*5^13 + 1*5^14 + 3*5^15 + 2*5^16 + 4*5^17 + 4*5^19 + ...

References

  • Kurt Mahler, Introduction to p-adic numbers and their functions. Cambridge Tracts in Mathematics, 76. Cambridge University Press, Cambridge-New York, 1971. See pp. 35ff.

Crossrefs

Cf. A034935, A048898, A048899, A210850, A210851.

Programs

  • Maple
    R:= select(t -> padic:-ratvaluep(t,1)=2,[padic:-rootp(x^2+1,5,200)]):
subs(0=NULL,op([1,1,3],R)); # Robert Israel, Mar 04 2016
  • PARI
    sqrt(-1+O(5^100))

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com) and Jason Earls, Jun 15 2001
Name corrected by Robert Israel at the suggestion of Wolfdieter Lang, Mar 04 2016
Previous Showing 21-25 of 25 results.

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