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 1-4 of 4 results.

A359282 Decimal expansion of Integral_{x = 0..1} 1/x^(x^2) dx.

Original entry on oeis.org

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

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			1.119545120136127596612676247029827036460046957876427628986749 ...

Links

Crossrefs

Cf. A073009, A083648, A253299, A359283, A359284, A359285, A359286.

Programs

  • Maple
    evalf(Sum(1/(2*n-1)^n, n = 1..infinity), 120);
  • Mathematica
    NIntegrate[x^(-x^2), {x, 0, 1}, WorkingPrecision -> 103] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(-x^2))

Formula

Equals Sum_{n >= 1} 1/(2*n - 1)^n.
More generally, Integral_{x = 0..1} 1/x^(t*x^2) dx = Sum_{n >= 1} t^(n-1)/(2*n - 1)^n. See A253299 (case t = -1).

A359284 Decimal expansion of Integral_{x = 0..1} 1/x^(x^3) dx.

Original entry on oeis.org

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

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			1.06551820592764917586382140548454723153980227909982...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359285, A359286.

Programs

  • Maple
    evalf(int(1/x^(x^3), x = 0..1), 110);
  • Mathematica
    NIntegrate[1/x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(-x^3))

Formula

Equals Sum_{n >= 1} 1/(3*n - 2)^n.
More generally, Integral_{x = 0..1} 1/x^(t*x^3) dx = Sum_{n >= 1} t^(n-1)/(3*n - 2)^n. See A359285 (case t = -1).

A359285 Decimal expansion of Integral_{x = 0..1} x^(x^3) dx.

Original entry on oeis.org

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

Views

Author

Peter Bala, Dec 24 2022

Keywords

Examples

			0.94031808668190698289736564174297672581175110149306...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359284, A359286.

Programs

  • Maple
    evalf(int(x^(x^3), x = 0..1), 110);
  • Mathematica
    NIntegrate[x^(x^3), {x, 0, 1}, WorkingPrecision -> 110] // RealDigits // First
  • PARI
    intnum(x = 0, 1, x^(x^3))

Formula

Equals Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n.

A359286 Decimal expansion of Integral_{x = 1..oo} 1/x^(x^3) dx.

Original entry on oeis.org

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

Views

Author

Peter Bala, Dec 24 2022

Keywords

Comments

For a, b nonnegative integers, the alternating divergent series Sum_{n >= 0} (-1)^n*(a*n + b)^n is Borel summable to Integral_{x = 1..oo} x^(a-b-1)/x^(x^a) dx.

Examples

			0.35854271600033996570705760779181131168203620572130...

Links

Crossrefs

Cf. A245637, A253299, A359282, A359283, A359284, A359285.

Programs

  • Maple
    evalf(int(1/x^(x^3), x = 1..infinity), 110);
  • Mathematica
    NIntegrate[1/x^(x^3), {x, 1, Infinity}, WorkingPrecision -> 110] // RealDigits // First

Formula

Equals Integral_{x = 1..oo} 1/(3*x - 2)^x dx.
Equals the Borel sum of the alternating divergent series Sum_{n >= 0} (-1)^n*(3*n + 2)^n. Compare with the alternating convergent series Sum_{n >= 1} (-1)^(n+1)/(3*n - 2)^n = Integral_{x = 0..1} x^(x^3) dx. See A359285.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.