A083648 -id:A083648 - cp's OEIS frontend

A322009 1/(Integral_{x=0..1} x^(x^(x^n)) dx - 1/2), rounded to the nearest integer.

4, 14, 33, 64, 110, 174, 260, 369, 506, 672, 872, 1108, 1382, 1699, 2061, 2472, 2933, 3448, 4021, 4653, 5349, 6110, 6941, 7844, 8822, 9878, 11015, 12237, 13545, 14943, 16435, 18023, 19709, 21498, 23392, 25394, 27507, 29734, 32079, 34543, 37131, 39844, 42687, 45662, 48772

Offset: 0

M. F. Hasler, Mar 01 2019

nonn

Linked to the problem of sorting parenthesized expressions (x^x....^x) (cf. A000081 and A222379, A222380) according to the value of their integral from 0 to 1: This value is minimal, for a given number n of x's, for G[n](x) := x^((...(x^x)^x....)^x) = x^(x^(x^(n-2))), which converges pointwise to x^(x^0) = x^1 = x for all x in [0,1], as n -> oo. The corresponding integrals therefore tend to 1/2 as n -> oo. This sequence is a convenient measure of the distance of these integrals from 1/2.

See A322008 for the maximal values of such integrals.

			For n=0, Integral_{x=0..1} x^(x^(x^0)) dx = Integral_{x=0..1} x^x dx = A083648 = 0.7834..., and 1/(0.7834... - 0.5) = 1 / 0.2834... = 3.528..., so a(0) = round(3.528...) = 4.
For n=1, Integral_{x=0..1} x^(x^(x^1)) dx = Integral_{x=0..1} x^(x^x) dx = 0.5731..., and 1/(0.5731... - 0.5) = 1 / 0.0731... = 13.67..., so a(1) = round(13.67...) = 14.

Vladimir Reshetnikov, Integrals of power towers, on MathOverflow.net, Feb. 26, 2019.

Cf. A322008; A000081, A222379, A222380; A083648.

Digits:= 20:
a:= n-> round(evalf(1/(int(x^(x^(x^n)), x=0..1)-1/2))):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 01 2019

f[n_] := Round[1/(NIntegrate[x^(x^(x^n)), {x, 0, 1}, WorkingPrecision -> 24] - 1/2)]; Array[f, 45, 0] (* Robert G. Wilson v, Mar 01 2019 *)

PARI
```
A322009(n)=1\/intnum(x=0, 1, x^x^x^n-x)
```

A347345 Decimal expansion of 1 / 1^1 + 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) + 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 28 2021

This constant is irrational (Stucky, 2016). - Amiram Eldar, Apr 30 2023

			1.2592954398150628876913596498837350926311457518421...

Joshua Stucky, Several Theorems Concerning the Irrationality of Certain Infinite Series Involving Tetrations, The Pentagon, Vol. 75, No. 2 (2016), pp. 4-9.
Eric Weisstein's World of Mathematics, Hyperfactorial.

Cf. A000312, A002109, A073009, A083648, A091131, A137986, A287013, A347352.

With[{m = 105}, RealDigits[N[Sum[1/Hyperfactorial[n], {n, 1, Infinity}], m + 2], 10, m][[1]]] (* Amiram Eldar, Apr 30 2023 *)
RealDigits[Total[Table[1/Times@@(Range[n]^Range[n]),{n,30}]],10,120][[1]] (* Harvey P. Dale, Oct 07 2023 *)

A347352 Decimal expansion of 1 / 1^1 - 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) - 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 28 2021

			0.759223101851603779577759044955845595913352402184...

Eric Weisstein's World of Mathematics, Hyperfactorial.

Cf. A000312, A002109, A068996, A073009, A083648, A137986, A287013, A347345.

With[{m = 105}, RealDigits[N[Sum[(-1)^(n + 1)/Hyperfactorial[n], {n, 1, Infinity}], m + 2], 10, m][[1]]] (* Amiram Eldar, Apr 30 2023 *)

A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

Original entry on oeis.org

1, 1, 1, 4, 2, 2, 27, 9, 6, 6, 256, 64, 32, 24, 24, 3125, 625, 250, 150, 120, 120, 46656, 7776, 2592, 1296, 864, 720, 720, 823543, 117649, 33614, 14406, 8232, 5880, 5040, 5040, 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320

Offset: 0

Robert B Fowler, Dec 27 2021

T(n,k) are the denominators in a double summation power series for the definite integral of x^x. First expand x^x = exp(x*log(x)) = Sum_{n>=0} (x*log(x))^n/n!, then integrate each of the terms to get the double summation for F(x) = Integral_{t=0..x} t^t = Sum_{n>=1} (Sum_{k=0..n-1} (-1)^(n+k+1)*x^n*(log(x))^k/T(n,k)).
This is a definite integral, because lim {x->0} F(x) = 0.
The value of F(1) = 0.78343... = A083648 is known humorously as the Sophomore's Dream (see Borwein et al.).

			Triangle T(n,k) begins:
--------------------------------------------------------------------------
n/k         0        1       2       3      4      5      6      7      8
--------------------------------------------------------------------------
0  |        1,
1  |        1,       1,
2  |        4,       2,      2,
3  |       27,       9,      6,      6,
4  |      256,      64,     32,     24,    24,
5  |     3125,     625,    250,    150,   120,   120,
6  |    46656,    7776,   2592,   1296,   864,   720,   720,
7  |   823543,  117649,  33614,  14406,  8232,  5880,  5040,  5040,
8  | 16777216, 2097152, 524288, 196608, 98304, 61440, 46080, 40320, 40320.
...

Borwein, J., Bailey, D. and Girgensohn, R., Experimentation in Mathematics: Computational Paths to Discovery, A. K. Peters 2004.
William Dunham, The Calculus Gallery, Masterpieces from Newton to Lebesgue, Princeton University Press, Princeton NJ 2005.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Eric Weisstein's World of Mathematics, Sophomore's dream
Wikipedia, Sophomore's dream

Cf. A000312 (first column), A000169 (2nd column), A003308 (3rd column excluding first term), A000142 (main diagonal), A000142 (2nd diagonal excluding first term), A112541 (row sums).
Values of the integral: A083648, A073009.
Cf. A061711, A350297.

A350149:= func< n,k | n^(n-k)*Factorial(k) >;
[A350149(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 31 2022

T := (n, k) -> n^(n - k)*k!:
seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jan 07 2022

T[n_, k_]:= n^(n-k)*k!; Table[T[n, k], {n, 0,12}, {k,0,n}]//Flatten (* Amiram Eldar, Dec 27 2021 *)

def A350149(n,k): return n^(n-k)*factorial(k)
flatten([[A350149(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 31 2022

T(n, 0) = A000312(n).
T(n, 1) = A000169(n).
T(n, 2) = A003308(n), n >= 2.
Sum_{k=0..n} T(n, k) = A112541(n).
T(n, n) = A000142(n).
T(n, n-1) = A000142(n), n >= 1.
T(n,k) = A061711(n) * (n+1) / A350297(n+1,k). - Robert B Fowler, Jan 11 2022

A137420 Continued fraction expansion of Sum_{n=1..infinity} (-1)^(n+1)/n^n = Integral_{x=0..1} x^(x) dx.

Original entry on oeis.org

0, 1, 3, 1, 1, 1, 1, 1, 1, 2, 4, 7, 2, 1, 2, 1, 1, 1, 2, 1, 14, 1, 1, 2, 4, 1, 120, 1, 3, 1, 4, 1, 2, 6, 1, 1, 5, 1, 1, 5, 2, 1, 11, 2, 409, 1, 1, 7, 3, 2, 1, 11, 142, 1, 3, 1, 44, 1, 1, 27, 1, 3, 1, 1, 100, 1, 39, 14, 2, 16, 1, 1, 11, 1, 2, 29, 2, 1, 1, 1, 3, 4, 1, 1, 1, 12, 1, 1, 118, 7, 9, 1, 1, 6

Offset: 0

Jani Melik, Apr 16 2008

Wikipedia, Sophomore's dream.

Cf. A083648 (decimal expansion).

sd1 := proc(n) local i, tren; tren := 0: for i from 1 to n do tren := (-1)^(i+1)*(1/i^(i)) + tren: od; RETURN(tren); end: numtheory[cfrac] (sd1(300),150,'quotients');

Offset changed by Andrew Howroyd, Aug 03 2024

A209059 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} (xyz)^(xyz) dx dy dz.

Original entry on oeis.org

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

Offset: 0

Peter Bala, Mar 04 2012

The double integral Integral_{y = 0..1} Integral_{x = 0..1} (x*y)^(x*y) dx dy equals Integral_{x = 0..1} x^x dx, which is listed as A083648.

			0.83493011063622359351...

Cf. A073009, A083648, A209060.

digits = 103; 1/2*NSum[ (-1)^(n+1)*(1/n^n + 1/n^(n+1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 100] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013, from formula *)

The triple integral is most conveniently estimated from the identity Integral_{z = 0..1} Integral_{y = 0..1} Integral_{z = 0..1} (x*y*z)^(x*y*z) dx dy dz = (1/2)*Sum_{n >= 1} (-1)^(n+1)*(1/n^n + 1/n^(n+1)).

A209060 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(xyz)^(xyz) dx dy dz.

Original entry on oeis.org

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

Offset: 1

Peter Bala, Mar 04 2012

Cf. A209059. The double integral Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y)^(x*y) dx dy equals Integral_{x = 0..1} 1/x^x dx, which is listed as A073009.

			1.21483799601716270069...

Cf. A073009, A083648, A135608, A209059.

digits = 103; 1/2*NSum[ (1/n^n + 1/n^(n+1)), {n, 1, Infinity}, WorkingPrecision -> digits+10, NSumTerms -> 100] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 15 2013, from formula *)

default( realprecision, 105); v = Vec( Str( suminf( n=1, n^-n + n^-(n+1)) / 20)); for( n=3, 105, print1( v[n],",")); /* Michael Somos, Mar 07 2012 */

The triple integral is most conveniently estimated from the identity Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz = 1/2*Sum_{n = 1..oo} (1/n^n + 1/n^(n+1)).

A253300 Decimal expansion of integral_{x=0..1} x^sqrt(x) dx.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Dec 30 2014

			0.6585823541090935654696568534036441701564...

Paul J. Nahin, Inside Interesting Integrals, Springer 2014, ISBN 978-1493912766.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.1.6)

Cf. A073009, A083648, A253299.

NIntegrate[x^Sqrt[x], {x, 0, 1}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First

intnum(x=0,1, x^sqrt(x)) \\ Michel Marcus, Dec 30 2014

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

A262974 Decimal expansion of Sum_{n > 0} (-1/n)^(n-1).

Original entry on oeis.org

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

Offset: 0

Andrew Penton, Oct 05 2015

			0.5969655555784832245791273664925691831645883...

Cf. A083648, A098686.

evalf(Sum((-1/n)^(n-1), n=1..infinity), 120); # Vaclav Kotesovec, Oct 15 2015

Mathematica
```
Sum[(-1/n)^(n-1),{n,1,Infinity}]
```

sumalt(n=1, (-1/n)^(n-1)) \\ Michel Marcus, Oct 06 2015

Equals 1 - Integral_{x = 0..1} x^(x+1) dx. - Peter Bala, Jul 21 2022

More terms from Michel Marcus, Oct 15 2015

A279020 a(n) = unreduced numerator in Sum_{k=1..n} (-1)^(k-1)/k^k.

Original entry on oeis.org

0, 1, 3, 85, 21652, 67690148, 3158065145088, 2600806474859606784, 43634288683151793919033344, 16904817514490272003065681518985216, 169048175123324778807714958980684133171200000, 48231417258392463006590622244366007983063473271603200000

Offset: 0

Daniel Suteu, Dec 03 2016

Daniel Suteu, Table of n, a(n) for n = 0..19
Wikipedia, Sophomore's dream.

Cf. A083648.

sum(k=1, n, (-1)^(k-1)/k^k) * prod(k=1, n, k^k)

a(n) ~ A083648 * A002109(n).

a(0) = 0, a(n) = n^n * a(n-1) + (-1)^(n-1) * A002109(n-1).

cp's OEIS Frontend

A322009 1/(Integral_{x=0..1} x^(x^(x^n)) dx - 1/2), rounded to the nearest integer.

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A347345 Decimal expansion of 1 / 1^1 + 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) + 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...

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A347352 Decimal expansion of 1 / 1^1 - 1 / (1^1 * 2^2) + 1 / (1^1 * 2^2 * 3^3) - 1 / (1^1 * 2^2 * 3^3 * 4^4) + ...

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A350149 Triangle read by rows: T(n, k) = n^(n-k)*k!.

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A137420 Continued fraction expansion of Sum_{n=1..infinity} (-1)^(n+1)/n^n = Integral_{x=0..1} x^(x) dx.

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A209059 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} (x*y*z)^(x*y*z) dx dy dz.

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A209060 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(x*y*z)^(x*y*z) dx dy dz.

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A253300 Decimal expansion of integral_{x=0..1} x^sqrt(x) dx.

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A209059 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} (xyz)^(xyz) dx dy dz.

A209060 Decimal expansion of the triple integral Integral_{z = 0..1} Integral_{y = 0..1} Integral_{x = 0..1} 1/(xyz)^(xyz) dx dy dz.