A368617 -id:A368617 - cp's OEIS frontend

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.

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

A368620 a(n) is the n-digit numerator 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

3, 87, 878, 2721, 49171, 566827, 9242691, 28245729, 848456353

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                 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, A368621 (denominator), A368659.

Cf. A007676, A007677, A368618, A368619.

a[1]=3; 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; hmin = i]]]; hmin]; Array[a,9]

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]

A368653 Decimal expansion of 58291/21444.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 02 2024

It is a rational approximation of e having an error less than 0.0003% provided by Charles Hermite in 1874 (see Hermite and Maor), where the error is calculated by abs(58291/21444-e)/e and expressed in percent.

Periodic with a period length of 893. - Ray Chandler, Jan 19 2024

			2.7182894982279425480320835664987875396381272...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 189.

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.
Charles Hermite, Sur la fonction exponentielle, Gauthier-Villars, Paris, 1874.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, order 893.

Cf. A001113, A007676, A007677, A368617, A368654, A368656.

Flatten[First[RealDigits[58291/21444,10,100]]]

A368654 Decimal expansion of 158452/21444.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Jan 02 2024

It is a rational approximation of e^2 (A072334) provided by Charles Hermite in 1874 (see Hermite and Maor).

Periodic with a period length of 893. - Ray Chandler, Jan 19 2024

			7.38910650997948144002984517813840701361686...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 189.

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.
Charles Hermite, Sur la fonction exponentielle, Gauthier-Villars, Paris, 1874.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, order 893.

Cf. A001113, A007676, A007677, A072334, A368617, A368653, A368656.

Flatten[First[RealDigits[158452/21444,10,100]]]

A368656 Decimal expansion of 271801/99990.

Original entry on oeis.org

2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8, 1, 8, 2, 8

Offset: 1

Stefano Spezia, Jan 02 2024

Periodic with a period length of 4. - Ray Chandler, Jan 19 2024

			2.718281828182818281828182818281828182818281828...

Eli Maor, e: The Story of a Number. Princeton, New Jersey: Princeton University Press (1994), p. 37.

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. S30.
Index entries for sequences related to the number e
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001113, A007676, A007677, A368617, A368653, A368654.

Flatten[First[RealDigits[271801/99990,10,100]]]

G.f.: x*(-x^5 - 8*x^3 - x^2 - 7*x - 2)/(x^4 - 1). - Chai Wah Wu, Jan 04 2024

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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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A368620 a(n) is the n-digit numerator of the fraction h/k with h and k coprime 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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A368653 Decimal expansion of 58291/21444.

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A368654 Decimal expansion of 158452/21444.

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A368656 Decimal expansion of 271801/99990.

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