A073009 -id:A073009 - cp's OEIS frontend

A253299 Decimal expansion of integral_{x=0..1} x^(x^2) dx.

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

Offset: 0

Jean-François Alcover, Dec 30 2014

			0.896488781929623341300238520792550365918625...

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

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.1.5)

Cf. A073009, A083648, A253300.

NIntegrate[x^(x^2), {x, 0, 1}, WorkingPrecision -> 103] // RealDigits // First

intnum(x=0,1, x^(x^2)) \\ Michel Marcus, Dec 30 2014

Equals sum_{n >= 0} (-1)^n/(2n + 1)^(n + 1).

A134878 Decimal expansion of Sum_{k>=1} 1/(k^2)^(k^2).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 14 2007

Sum_{k>=1} 1/(k^2)^(k^2) = 1.003906252581174791767407290614306741076124924379285...

Cf. A073009, A100085, A134877.

RealDigits[N[Sum[1/(n^2)^(n^2), {n, 1, 30}], 200]][[1]]

Equals Sum_{n>=1} 1/A008972(n). - R. J. Mathar, Jul 31 2025

A134879 Decimal expansion of Sum_{k>=1} 1/(k^3)^(k^3).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Nov 14 2007

Sum_{k>=1} 1/(k^3)^(k^3) = 1.00000005960464477539062500000000000000225516523372...

Cf. A073009, A100085, A134877, A134878.

RealDigits[N[Sum[1/(n^3)^(n^3), {n, 1, 30}], 200]][[1]]

suminf(n=1,n^(-3*n^3)) \\ Charles R Greathouse IV, Dec 26 2011

A308915 Decimal expansion of Sum_{n>=1} 1/(log(n)^log(n)).

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Jun 30 2019

This series is convergent because n^2 * 1/log(n)^log(n) = exp(log(n) * (2 - log(log(n)))) which -> 0 as n -> oo.

			6.71697061299089608814457...

Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.1.i p. 279.

Cf. A073009 (1/n^n), A099870 (1/n^log(n)), A099871 (1/log(n)^n).

evalf(sum(1/(log(n)^log(n)), n=1..infinity), 110);

RealDigits[N[1 + Sum[1/Log[n]^Log[n], {n, 2, Infinity}], 100]][[1]] (* Jinyuan Wang, Jul 25 2019 *)

1 + sumpos(n=2, 1/(log(n)^log(n))) \\ Michel Marcus, Jun 30 2019

Equals Sum_{n>=1} 1/(log(n)^log(n)).

More terms from Jon E. Schoenfield, Jun 30 2019
a(16)-a(24) from Jinyuan Wang, Jul 10 2019
More terms from Charles R Greathouse IV, Oct 21 2021

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

Peter Bala, Dec 24 2022

			1.119545120136127596612676247029827036460046957876427628986749 ...

Wikipedia, Sophomore's Dream

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

evalf(Sum(1/(2*n-1)^n, n = 1..infinity), 120);

NIntegrate[x^(-x^2), {x, 0, 1}, WorkingPrecision -> 103] // RealDigits // First

PARI
```
intnum(x = 0, 1, x^(-x^2))
```

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).

A061464 Denominator of 1/(1^1) + 1/(2^2) + 1/(3^3) + ... 1/(n^n).

Original entry on oeis.org

1, 1, 4, 108, 6912, 21600000, 583200000, 480290277600000, 31476303632793600000, 16727798278915463577600000, 52274369621610823680000000000, 14914487726878692033020558868480000000000

Offset: 0

Amarnath Murthy, May 04 2001

			1, 5/4, 139/108, 8923/6912,...

Cf. A061463, A073009.

summ := 0; for n from 1 to 15 do printf("%d ", denom(summ)); if (1 = 1) then summ := summ + 1/n^n: end if; od;

Join[{1},Denominator/@Table[Sum[1/i^i,{i,n}],{n,12}]] (* Harvey P. Dale, Jul 03 2011 *)

a(n) = denominator(sum(k=1, n, 1/(k^k))) \\ Thomas Scheuerle, Feb 26 2025

A061463(n)/a(n) = Integral_{x=0..1} Gamma(n, -x*log(x))/(x^x*Gamma(n)) dx. - Thomas Scheuerle, Feb 26 2025

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), May 19 2001

a(12) from Harvey P. Dale, Jul 03 2011

A062071 a(n) = [n/1] + [n/(2^2)] + [n/(3^3)] + [n/(4^4)] + ... + [n/(k^k)] + ..., up to infinity, where [ ] is the floor function.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 87, 88, 89, 90

Offset: 1

Amarnath Murthy, Jun 13 2001

			a(7) = [7/1] + [7/4] + [7/27] + ... = 7 + 1 + 0 + 0 + ... = 8.
a(8) = [8/1] + [8/4] + [8/27] + [8/256] + ... = 8 + 2 + 0 + 0 + ... = 10.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Vaclav Kotesovec, Plot of a(n)/n for n = 1..100000

Cf. A006218, A011371, A060832, A347397.

Flatten[{1, Table[Sum[Floor[n/k^k], {k, 1, Floor[N[Log[n]/LambertW[Log[n]]]] + 1}], {n, 2, 100}]}] (* Vaclav Kotesovec, Aug 30 2021 *)

\p 10 v=[]; for(n=1,120,v=concat(v,suminf(k=1,floor(n/k^k)))); v

for (n=1, 1000, write("b062071.txt", n, " ", suminf(k=1, n\k^k)\1) ) \\ Harry J. Smith, Jul 31 2009

a(n)=sum(k=1,exp(lambertw(log(n)))+1,n\k^k) \\ Charles R Greathouse IV, May 28 2015

[sum( floor(n/j^j) for j in (1..1+log(n)) ) for n in (1..100)] # G. C. Greubel, May 06 2022

a(n) = Sum_{i=1..n} floor(n/i^i). - Wesley Ivan Hurt, Sep 15 2017
G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^k)/(1 - x^(k^k)). - Seiichi Manyama, Aug 30 2021
Conjecture: a(n) ~ c * n, where c = A073009. - Vaclav Kotesovec, Aug 30 2021

More terms from Jason Earls, Jun 21 2001

A085529 a(n) = (2n+1)^(2n+1).

Original entry on oeis.org

1, 27, 3125, 823543, 387420489, 285311670611, 302875106592253, 437893890380859375, 827240261886336764177, 1978419655660313589123979, 5842587018385982521381124421, 20880467999847912034355032910567, 88817841970012523233890533447265625, 443426488243037769948249630619149892803

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

a(n) == 2*n + 1 (mod 24). - Mathew Englander, Aug 16 2020

Cf. A000312, A005408, A016754, A085527, A085528, A085530, A085531, A085532, A085533, A085534, A085535.

Cf. A073009, A083648.

#^#&/@Range[1,31,2] (* Harvey P. Dale, Oct 05 2019 *)

From Mathew Englander, Aug 16 2020: (Start)
a(n) = A000312(2*n + 1).
a(n) = A016754(n)^n * (2*n + 1).
a(n) = A085527(n)^2 * (2*n + 1).
a(n) = A085528(n)^2 / (2*n + 1).
a(n) = A085530(n) * A005408(n).
a(n) = A085531(n) * A016754(n).
a(n) = A085532(n)^2 - A215265(2*n + 1).
a(n) = A085533(n) + A045531(2*n + 1).
a(n) = A085534(n+1) - A007781(2*n + 1).
a(n) = A085535(n+1) - A055869(2*n + 1).
(End)
Sum_{n>=0} 1/a(n) = (A073009 + A083648)/2 = 1.0373582538... . - Amiram Eldar, May 17 2022

A086648 Decimal expansion of the sum n^(-2n) for n=1 through infinity.

Original entry on oeis.org

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

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Jul 26 2003

This number is the sum of the inverses of the terms in the sequence A062206.

			1.063887103762417011746491117660011115727199076891599286634083869667...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A062206, A073009.

suminf(k=1,1./k^(2*k)) \\ Franklin T. Adams-Watters, Mar 23 2010

Clarified definition - R. J. Mathar, Feb 06 2009

More terms from Franklin T. Adams-Watters, Mar 23 2010

A134880 Decimal expansion of Sum_{k>=1} 1/(2^k)^(2^k).

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Nov 14 2007

			0.25390630960464477544483510862427522170037264004418131333707266458541197733559...

Cf. A073009, A100085, A134877, A134878, A134879.

RealDigits[N[Sum[1/(2^n)^(2^n), {n, 1, 30}], 200]][[1]]

suminf(k=1, 1/(2^k)^(2^k)) \\ Michel Marcus, Jan 15 2021

cp's OEIS Frontend

A253299 Decimal expansion of integral_{x=0..1} x^(x^2) dx.

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A134878 Decimal expansion of Sum_{k>=1} 1/(k^2)^(k^2).

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A134879 Decimal expansion of Sum_{k>=1} 1/(k^3)^(k^3).

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A308915 Decimal expansion of Sum_{n>=1} 1/(log(n)^log(n)).

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

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A061464 Denominator of 1/(1^1) + 1/(2^2) + 1/(3^3) + ... 1/(n^n).

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A062071 a(n) = [n/1] + [n/(2^2)] + [n/(3^3)] + [n/(4^4)] + ... + [n/(k^k)] + ..., up to infinity, where [ ] is the floor function.

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