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 1-5 of 5 results.

A356670 a(n) is the number of correct decimal digits of Pi obtained from the fraction A355622(n)/A355623(n).

Original entry on oeis.org

0, 2, 3, 2, 4, 4, 7, 8, 8, 10, 10, 11, 11, 14, 14, 15, 16, 18, 18, 18, 19, 20, 22
Offset: 1

Views

Author

Stefano Spezia, Aug 22 2022

Keywords

Examples

			a(9) = 8 since A355622(9)/A355623(9) = 422334431/134433224 = 3.1415926690860... matches the first 8 digits of Pi: 3.1415926535897...

Crossrefs

Cf. A000796, A355622, A355623, A356671.

Extensions

a(19)-a(22) from Stefano Spezia, Oct 13 2022

A355622 a(n) is the n-digit positive number with no trailing zeros and coprime to its digital reversal R(a(n)) at which abs(a(n)/R(a(n))-Pi) is minimized.

Original entry on oeis.org

1, 92, 581, 5471, 52861, 998713, 7774742, 93630892, 422334431, 9190135292, 45425395441, 472539314051, 5784475521481, 49371008251751, 939253175379892, 9265811239939492, 52949745472445861, 952186420153090303, 9836241210282790313, 36386277546811128511, 442327789252803797041
Offset: 1

Views

Author

Stefano Spezia, Jul 10 2022

Keywords

Comments

a(n) and R(a(n)) have the same number of digits.
Petr Beckmann wrote that the fraction 92/29, corresponding to the second term of the sequence, appeared as value of Pi in a document written in A.D. 718.

Examples

			n              fraction    approximated value
-   -------------------    ------------------
1                     1    1
2                 92/29    3.1724137931034...
3               581/185    3.1405405405405...
4             5471/1745    3.1352435530086...
5           52861/16825    3.1418127786033...
6         998713/317899    3.1416047235128...
7       7774742/2474777    3.1415929596889...
8     93630892/29803639    3.1415926088757...
9   422334431/134433224    3.1415926690860...
...

References

  • Petr Beckmann, A History of Pi, 3rd Ed., Boulder, Colorado: The Golem Press (1974): p. 27.

Links

Crossrefs

Cf. A000796, A002485, A004086, A067251.
Cf. A355623 (denominator or digital reversal).

Programs

  • Mathematica
    nmax=9; a={}; For[n=1, n<=nmax, n++, minim=Infinity; For[k=10^(n-1), k<=10^n-1, k++, If[(dist=Abs[k/FromDigits[Reverse[IntegerDigits[k]]]-Pi]) < minim && Last[IntegerDigits[k]]!=0 && GCD[k,FromDigits[Reverse[IntegerDigits[k]]]]==1, minim=dist; kmin=k]]; AppendTo[a, kmin]]; a

Extensions

a(10)-a(19) from Bert Dobbelaere, Jul 17 2022
a(20)-a(21) from Bert Dobbelaere, Sep 05 2022

A364844 a(n) is the n-digit numerator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal.

Original entry on oeis.org

3, 22, 474, 1551, 36163, 292292, 7327237, 31311313
Offset: 1

Views

Author

Stefano Spezia, Aug 10 2023

Keywords

Comments

a(2) = 22 corresponds to the numerator of A068028.

Examples

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                     3    3
  2                  22/7    3.1428571428571...
  3               474/151    3.1390728476821...
  4              1551/494    3.1396761133603...
  5           36163/11511    3.1416036834332...
  6          292292/93039    3.1416072829673...
  7       7327237/2332332    3.1415926206046...
  8      31311313/9966699    3.1415931192464...
  ...

Links

Crossrefs

Cf. A000796, A002113, A068028, A070252, A364845 (denominator), A364846.
Cf. A355622, A355623.

Programs

  • Mathematica
    nmax = 8; a = {3}; hmin = kmin = 0; For[n = 2, n <= nmax, n++, minim = Infinity; h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; k = Select[Range[10^(n - 2), 10^n - 1], PalindromeQ]; lh = Length[h]; lk = Length[k]; For[i = 1, i <= lh, i++, For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - Pi]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; hmin = Part[h, i]]]]; AppendTo[a, hmin]]; a

A364845 a(n) is the denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-Pi) is minimal, with the numerator h of n digits.

Original entry on oeis.org

1, 7, 151, 494, 11511, 93039, 2332332, 9966699
Offset: 1

Views

Author

Stefano Spezia, Aug 10 2023

Keywords

Comments

a(2) = 7 corresponds to the denominator of A068028.

Examples

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                     3    3
  2                  22/7    3.1428571428571...
  3               474/151    3.1390728476821...
  4              1551/494    3.1396761133603...
  5           36163/11511    3.1416036834332...
  6          292292/93039    3.1416072829673...
  7       7327237/2332332    3.1415926206046...
  8      31311313/9966699    3.1415931192464...
  ...

Links

Crossrefs

Cf. A000796, A002113, A068028, A070252, A364844 (numerator), A364846.
Cf. A355622, A355623.

Programs

  • Mathematica
    nmax = 8; a = {1}; hmin = kmin = 0; For[n = 2, n <= nmax, n++, minim = Infinity; h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; k = Select[Range[10^(n - 2), 10^n - 1], PalindromeQ]; lh = Length[h]; lk = Length[k];  For[i = 1, i <= lh, i++, For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - Pi]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; kmin = Part[k, j]]]]; AppendTo[a, kmin]]; a

A380100 a(n) is the denominator of the fraction h/k with h and k coprime positive integers at which abs((h/k)^4-Pi) is minimal, with the numerator h of n digits.

Original entry on oeis.org

3, 73, 667, 7174, 10177, 379552, 3456676, 66066573, 223935013
Offset: 1

Views

Author

Stefano Spezia, Jan 12 2025

Keywords

Comments

a(1)^4 = 3^4 = 81 corresponds to the denominator of A210621.
It appears that the number of correct decimal digits of Pi obtained from the fraction A380099(n)/a(n) is A130773(n-1) for n > 1 (see Spezia in Links). - Stefano Spezia, Apr 20 2025

Examples

			  n               (h/k)^4    approximated value
  -   -------------------    ------------------
  1               (4/3)^4    3.1604938271604...
  2             (97/73)^4    3.1174212867620...
  3           (888/667)^4    3.1415829223858...
  4         (9551/7174)^4    3.1415927852873...
  5       (13549/10177)^4    3.1415926560044...
  ...

Links

Crossrefs

Cf. A000583, A000796, A130773, A210621.
Cf. A355623, A364845, A380099 (numerator).

Programs

  • Mathematica
    nmax = 3; a = {}; hmin = kmin = 0; For[n = 1, n <= nmax, n++, minim = Infinity; For[h = 10^(n-1), h <10^n, h++, For[k = 1, k < 10^n/Pi^(1/4), k++, If[(dist = Abs[h^4/k^4-Pi]) < minim && GCD[h,k]==1, minim = dist; hmin=h; kmin = k]]]; AppendTo[a, kmin]]; a

Extensions

a(6)-a(9) from Kritsada Moomuang, Apr 17 2025
Showing 1-5 of 5 results.

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