A368620 -id:A368620 - cp's OEIS frontend

A368659 a(n) is the number of correct decimal digits of e obtained from the fraction A368620(n)/A368621(n).

0, 4, 5, 7, 10, 11, 13, 14, 18

Offset: 1

Stefano Spezia, Jan 02 2024

			a(8) = 14 since A368620(8)/A368621(8) = 28245729/10391023 = 2.7182818284590... matches the first 14 digits of e: 2.71828182845904523...

Index entries for sequences related to the number e

Cf. A001113, A368620, A368621, A368658.

A368617 Decimal expansion of 878/323.

Original entry on oeis.org

2, 7, 1, 8, 2, 6, 6, 2, 5, 3, 8, 6, 9, 9, 6, 9, 0, 4, 0, 2, 4, 7, 6, 7, 8, 0, 1, 8, 5, 7, 5, 8, 5, 1, 3, 9, 3, 1, 8, 8, 8, 5, 4, 4, 8, 9, 1, 6, 4, 0, 8, 6, 6, 8, 7, 3, 0, 6, 5, 0, 1, 5, 4, 7, 9, 8, 7, 6, 1, 6, 0, 9, 9, 0, 7, 1, 2, 0, 7, 4, 3, 0, 3, 4, 0, 5, 5, 7

Offset: 1

Stefano Spezia, Jan 01 2024

Using the criterion of the minimum absolute difference, it is the closest rational approximation of e (cf. A001113) using integers below 1000 (see Maor).
The numerator and the denominator of 878/323 are palindromes in base 10 (cf. A002113).
It has period 144.

			2.7182662538699690402476780...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 37.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 46.

Philip R. Brown, Approximations of e and Pi: an exploration, International Journal of Mathematical Education in Science and Technology, 48:sup1, S30-S39, 2017. See p. S32.
Index entries for sequences related to the number e.
Index entries for linear recurrences with constant coefficients, order 143.

Cf. A001113, A002113, A007676, A007677, A368653, A368654, A368656.

Cf. A368618, A368619, A368620, A368621.

Mathematica
```
Flatten[First[RealDigits[878/323,10]]]
```

Equals A368618(3)/A368619(3) = A368620(3)/A368621(3).

A368618 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-e) is minimal.

Original entry on oeis.org

3, 11, 878, 2552, 38983, 167761, 4407044, 24988942, 882646288, 1385885831, 83034443038, 161356653161, 9051164611509, 24911822811942

Offset: 1

Stefano Spezia, Jan 01 2024

a(3) = 878 corresponds to the numerator of A368617.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                   3/1    3
  2                  11/4    2.75
  3               878/323    2.7182662538699...
  4              2552/939    2.7177848775292...
  5           38983/14341    2.7182902168607...
  6          167761/61716    2.7182740294251...
  7       4407044/1621261    2.7182816338640...
  8      24988942/9192919    2.7182815382143...
  9   882646288/324707423    2.7182818299783...
  ...

David A. Corneth, PARI program
Index entries for sequences related to the number e

Cf. A001113, A002113, A070252, A368617, A368619 (denominator), A368658.
Cf. A007676, A007677.
Cf. A364844 (similar for Pi), A368620, A368621.

a[1]=3; a[n_]:=Module[{minim = Infinity}, h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; lh = Length[h]; For[i = 1, i <= lh, i++, k = Select[Range[Floor[Part[h, i]/E], Ceiling[Part[h, i]/E]], PalindromeQ]; lk = Length[k]; For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - E]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; hmin = Part[h, i]]]]; hmin]; Array[a,9]

PARI
```
\\ See PARI program in Links
```

a(10)-a(14) from David A. Corneth, Jan 03 2024

A368619 a(n) is the n-digit denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-e) is minimal.

Original entry on oeis.org

1, 4, 323, 939, 14341, 61716, 1621261, 9192919, 324707423, 509838905, 30546664503, 59359795395, 3329737379233, 9164547454619

Offset: 1

Stefano Spezia, Jan 01 2024

a(3) = 323 corresponds to the denominator of A368617.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                   3/1    3
  2                  11/4    2.75
  3               878/323    2.7182662538699...
  4              2552/939    2.7177848775292...
  5           38983/14341    2.7182902168607...
  6          167761/61716    2.7182740294251...
  7       4407044/1621261    2.7182816338640...
  8      24988942/9192919    2.7182815382143...
  9   882646288/324707423    2.7182818299783...
  ...

David A. Corneth, PARI program
Index entries for sequences related to the number e

Cf. A001113, A002113, A070252, A368617, A368618 (numerator), A368658.
Cf. A007676, A007677.
Cf. A364845 (similar for Pi), A368620, A368621.

a[1]=1; a[n_]:=Module[{minim = Infinity}, h = Select[Range[10^(n - 1), 10^n - 1], PalindromeQ]; lh = Length[h]; For[i = 1, i <= lh, i++, k = Select[Range[Floor[Part[h, i]/E], Ceiling[Part[h, i]/E]], PalindromeQ]; lk = Length[k]; For[j = 1, j <= lk, j++, If[(dist = Abs[Part[h, i]/Part[k, j] - E]) < minim && GCD[Part[h, i], Part[k, j]] == 1, minim = dist; kmin = Part[k, j]]]]; kmin]; Array[a,9]

PARI
```
\\ See PARI program in Links
```

a(10)-a(14) from David A. Corneth, Jan 03 2024

A368621 a(n) is the n-digit denominator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

Original entry on oeis.org

1, 32, 323, 1001, 18089, 208524, 3400196, 10391023, 312129649

Offset: 1

Stefano Spezia, Jan 01 2024

a(3) = 323 corresponds to the denominator of A368617.

			  n              fraction    approximated value
  -   -------------------    ------------------
  1                   3/1    3
  2                 87/32    2.71875
  3               878/323    2.7182662538699...
  4             2721/1001    2.7182817182817...
  5           49171/18089    2.7182818287356...
  6         566827/208524    2.7182818284705...
  7       9242691/3400196    2.7182818284593...
  8     28245729/10391023    2.7182818284590...
  ...

Index entries for sequences related to the number e

Cf. A001113, A368617, A368620 (numerator), A368659.

Cf. A007676, A007677, A368618, A368619.

a[1]=1; a[n_]:=Module[{minim=Infinity},For[i = 10^(n - 1), i <= 10^n - 1, i++, For[j = Floor[i/E], j <= Ceiling[i/E], j++, If[(dist = Abs[i/j - E]) < minim && GCD[i, j] == 1, minim = dist; kmin = j]]]; kmin]; Array[a,9]

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A368659 a(n) is the number of correct decimal digits of e obtained from the fraction A368620(n)/A368621(n).

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A368617 Decimal expansion of 878/323.

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A368618 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-e) is minimal.

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A368619 a(n) is the n-digit denominator of the fraction h/k with h and k coprime palindrome positive integers at which abs(h/k-e) is minimal.

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A368621 a(n) is the n-digit denominator of the fraction h/k with h and k coprime positive integers at which abs(h/k-e) is minimal.

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