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 1-7 of 7 results.

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.

A324023 One of the two successive approximations up to 5^n for 5-adic integer sqrt(6). This is the 1 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 1, 16, 16, 516, 1766, 4891, 36141, 270516, 661141, 6520516, 35817391, 35817391, 768239266, 4430348641, 16637379891, 108190114266, 413365895516, 1939244801766, 9568639333016, 85862584645516, 371964879567391, 1802476354176766, 4186662145192391, 51870377965504891
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

For n > 0, a(n) is the unique solution to x^2 == 6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 1 modulo 5.
A324024 is the approximation (congruent to 4 mod 5) of another square root of 6 over the 5-adic field.

Examples

			16^2 = 256 = 10*5^2 + 6 = 2*5^3 + 6;
516^2 = 266256 = 426*5^4 + 6;
1766^2 = 3118756 = 998*5^5 + 6.

Links

Crossrefs

Cf. A048898, A048899, A324025, A324026.
Approximations of 5-adic square roots:
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
this sequence, A324024 (sqrt(6)).

Programs

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

Formula

For n > 0, a(n) = 5^n - A324024(n).
a(n) = A048898(n)*A324028(n) mod 5^n = A048899(n)*A324027(n) mod 5^n.

A324024 One of the two successive approximations up to 5^n for 5-adic integer sqrt(6). This is the 4 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 4, 9, 109, 109, 1359, 10734, 41984, 120109, 1291984, 3245109, 13010734, 208323234, 452463859, 1673166984, 13880198234, 44397776359, 349573557609, 1875452463859, 9504846995109, 9504846995109, 104872278635734, 581709436838859, 7734266809885734, 7734266809885734
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

For n > 0, a(n) is the unique solution to x^2 == 6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 1 modulo 5.
A324023 is the approximation (congruent to 4 mod 5) of another square root of 6 over the 5-adic field.

Examples

			9^2 = 81 = 3*5^2 + 6;
109^2 = 11881 = 95*5^3 + 6 = 19*5^4 + 6;
1359^2 = 1846881 = 591*5^5 + 6.

Links

Crossrefs

Cf. A048898, A048899, A324025, A324026.
Approximations of 5-adic square roots:
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, this sequence (sqrt(6)).

Programs

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

Formula

For n > 0, a(n) = 5^n - A324023(n).
a(n) = A048898(n)*A324027(n) mod 5^n = A048899(n)*A324028(n) mod 5^n.

A324027 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-6). This is the 2 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 2, 12, 37, 162, 1412, 10787, 42037, 354537, 1526412, 3479537, 3479537, 3479537, 247620162, 3909729537, 10013245162, 101565979537, 711917542037, 2237796448287, 13681888245162, 51828860901412, 337931155823287, 1291605472229537, 10828348636292037, 58512064456604537
Offset: 0

Views

Author

Jianing Song, Sep 07 2019

Keywords

Comments

For n > 0, a(n) is the unique solution to x^2 == -6 (mod 5^n) in the range [0, 5^n - 1] and congruent to 2 modulo 5.
A324028 is the approximation (congruent to 3 mod 5) of another square root of 6 over the 5-adic field.

Examples

			12^2 = 144 = 6*5^2 - 6;
37^2 = 1369 = 11*5^3 - 6;
162^2 = 26244 = 42*5^4 - 6.

Links

Crossrefs

Cf. A048898, A048899, A324029, A324030.
Approximations of 5-adic square roots:
this sequence, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

Programs

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

Formula

For n > 0, a(n) = 5^n - A324028(n).
a(n) = A048898(n)*A324023(n) mod 5^n = A048899(n)*A324024(n) mod 5^n.

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.

A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 4, 4, 79, 79, 79, 3204, 18829, 331329, 1112579, 1112579, 20643829, 118300079, 850721954, 3292128204, 27706190704, 149776503204, 302364393829, 1065303846954, 8694698378204, 46841671034454, 332943965956329, 332943965956329, 5101315547987579, 28943173458143829
Offset: 0

Views

Author

Jianing Song, Sep 16 2019

Keywords

Comments

a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^2 + 9 is divisible by 5^n.

Examples

			The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.

Links

Crossrefs

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
A327302, this sequence (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

Programs

  • Maple
    R:= [padic:-rootp(x^2+9,5,101)]:
R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)):
seq(padic:-ratvaluep(R,n),n=0..100); # Robert Israel, Jan 16 2023
  • PARI
    a(n) = truncate(-sqrt(-9+O(5^n)))

Formula

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

A327302 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 1 (mod 5) case (except for n = 0).

Original entry on oeis.org

0, 1, 21, 46, 546, 3046, 12421, 59296, 59296, 840546, 8653046, 28184296, 125840546, 369981171, 2811387421, 2811387421, 2811387421, 460575059296, 2749393418671, 10378787949921, 48525760606171, 143893192246796, 2051241825059296, 6819613407090546, 30661471317246796
Offset: 0

Views

Author

Jianing Song, Sep 16 2019

Keywords

Comments

a(n) is the unique number k in [1, 5^n] and congruent to 1 mod 5 such that k^2 + 9 is divisible by 5^n.

Examples

			The unique number k in {1, 6, 11, 16, 21} such that k^2 + 9 is divisible by 25 is k = 21, so a(2) = 21.
The unique number k in {21, 46, 71, 96, 121} such that k^2 + 9 is divisible by 125 is k = 46, so a(3) = 46.
The unique number k in {46, 171, 296, 421, 546} such that k^2 + 9 is divisible by 625 is k = 546, so a(4) = 546.

Links

Crossrefs

For the digits of sqrt(-9) see A327304 and A327305.
Approximations of 5-adic square roots:
this sequence, A327303 (sqrt(-9));
A324027, A324028 (sqrt(-6));
A268922, A269590 (sqrt(-4));
A048898, A048899 (sqrt(-1));
A324023, A324024 (sqrt(6)).

Programs

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

Formula

a(1) = 1; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.
For n > 0, a(n) = 5^n - A327303(n).
Showing 1-7 of 7 results.

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