A073226 -id:A073226 - cp's OEIS frontend

A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(ed(n))) - e^gamma log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

2, 74801040398884800, 224403121196654400, 3066842656354276800, 6133685312708553600, 9200527969062830400, 18401055938125660800, 131874234223233902400, 263748468446467804800, 395622702669701707200, 791245405339403414400, 6198089008491993412800, 12396178016983986825600

Offset: 1

Amiram Eldar, Aug 25 2019

Nicolas proved that f(n) reaches its maximum at n = 2^7 * (3#)^4 * 5# * (7#)^2 * 19# * 47# * 277# * 45439# ~ 8.0244105... * 10^19786 which is the last term of this sequence (prime(n)# = A002110(n) is the n-th primorial).

Amiram Eldar, Table of n, a(n) for n = 1..68
Jean-Louis Nicolas, Quelques inégalités effectives entre des fonctions arithmétiques usuelles, Functiones et Approximatio, Vol. 39, No. 2 (2008), pp. 315-334. See Theorem 1.2.

Cf. A000005, A000203, A073004, A073226, A217660.

Subsequence of A025487.

A331094 Decimal expansion of Pi^Pi - e^e.

Original entry on oeis.org

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

Offset: 2

Ahmad J. Masad, Jan 08 2020

			21.30789736572864758123039575006221176...

Cf. A073233, A073226, A331189.

RealDigits[Pi^Pi - Exp[E], 10, 100][[1]] (* Amiram Eldar, Jan 09 2020 *)

Pi^Pi - exp(exp(1)) \\ Michel Marcus, Jan 09 2020

A331189 Decimal expansion of Pi^Pi + e^e.

Original entry on oeis.org

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

Offset: 2

Ahmad J. Masad, Jan 11 2020

			51.61642184868717596075...

Cf. A000796, A001113, A019314, A073233, A073226, A331094.

RealDigits[Pi^Pi + Exp[E], 10, 100][[1]] (* Amiram Eldar, Jan 11 2020 *)

Equals A073233 + A073226.

A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

Original entry on oeis.org

4, 5, 7, 15, 2075

Offset: 0

Marco Ripà, Apr 08 2022

The common hyperoperation sequence is defined as follows: hyper-0 = zeration, hyper-1 = addition, hyper-2 = multiplication, hyper-3 = exponentiation, hyper-4 = tetration, and so on...
Thus e[0]e = e + 2, e[1]e = 2*e, e[2]e = e^2, e[3]e = e^e, and so on.
The fifth term of the twin sequence of the present one, floor(Pi[4]Pi), is much larger than 2075 and it is harder to calculate, while the integer part of e[4]Pi is 37149801960 (17.9 million times bigger than a(4)).

			For n = 3, a(3) = floor(e[3]e) = floor(e^e) = 15.

Hellmuth Kneser, Reelle analytische Lösungen der Gleichung phi(phi(x)) = e^x und verwandter Funktionalgleichungen, J. reine angew. Math. 187, 56-67 (1950).
Sheldon Levenstein (user sheldonison), New fatou.gp program, Jul 10 2015, updated Aug 14 2019.
William Paulsen, Tetration.
William Paulsen, Tetration for complex bases, Advances in Computational Mathematics, Vol. 45, No. 1 (2019), pp. 243-267; ResearchGate link.
Wikipedia, Hyperoperation

Cf. A001113, A019762, A072334, A073226, A351727, A352396.

a(n) = floor(e[n]e).

A096181 Floor (e^(n / log(n))).

Original entry on oeis.org

17, 15, 17, 22, 28, 36, 46, 60, 76, 98, 125, 158, 201, 254, 320, 403, 506, 634, 793, 989, 1233, 1533, 1904, 2360, 2922, 3612, 4459, 5498, 6771, 8328, 10231, 12556, 15393, 18851, 23063, 28189, 34423, 41998, 51195, 62353, 75883, 92274, 112119, 136131

Offset: 2

Robert G. Wilson v, Jun 19 2004

nonn

Cf. A000149, A073226.

Table[ Floor[ E^(n/Log[n])], {n, 2, 45}]

Definition corrected and cross-refs added by Franklin T. Adams-Watters, Jan 25 2010

A096194 Engel expansion of exp(e).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 13, 27, 45, 212, 388, 735, 3422, 3913, 11095, 18232, 20624, 54745, 426826, 617936, 3424833, 3494454, 3664162, 5238283, 5650352, 17373150, 168850488, 716822623, 729646247, 1226034011, 3976121167

Offset: 1

Gerald McGarvey, Jul 25 2004

nonn

Cf. A073226.

EngelExp[ A_, n_ ] := Join[ Array[ 1 &, Floor[ A ]], First@Transpose @ NestList[ {Ceiling[ 1/Expand[ #[[ 1 ]] #[[ 2 ]] - 1 ]], Expand[ #[[ 1 ]] #[[ 2 ]] - 1]} &, {Ceiling[ 1/(A - Floor[A]) ], A - Floor[A]}, n - 1 ]]; EngelExp[ N[E^E, 2^8], 26] (* Robert G. Wilson v, Jul 28 2008 *)

Edited and extended by Robert G. Wilson v, Jul 28 2004

A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

Original entry on oeis.org

0, 15, 6, 2, 13, 1, 3, 6, 2, 1, 1, 5, 1, 1, 1, 9, 4, 1, 1, 1, 6, 7, 1, 2, 4, 1, 2, 2, 24, 1, 2, 4, 56, 1, 1, 2, 4, 1, 75, 1, 5, 1, 2, 2, 1, 137, 2, 2, 97, 3, 16, 1, 1, 1, 1, 3, 5, 12, 1, 1, 2, 1, 53, 1, 2, 5, 3, 2, 4, 1, 2, 1, 39, 1, 2, 1, 4, 1, 11, 1, 5, 5, 1, 4, 1, 17, 12, 4, 82, 1, 4, 6, 25, 3, 2, 3, 39

Offset: 1

Jonathan Vos Post, Mar 16 2006

e^(-e) = (1/e)^e = 1/(e^e) = (reciprocal of A073226). e^(-e) = 0.0659880358453125370767901875... = 0 + 1/15+ 1/6+ 1/2+ 1/13+ 1/1+ 1/3+ 1/6+ 1/2+ ... See also: A073230 Decimal expansion of (1/e)^e. See also: A064107 Continued fraction quotients for e^e = 15.15426223. See also: A058287 Continued fraction for e^Pi. See also: A058288 Continued fraction expansion of Pi^e.

Cf. A064107, A058287, A058288, A073226.

A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

Original entry on oeis.org

1, 2, 3, 16, 3814280

Offset: 1

Alexis Monnerot-Dumaine, Jan 21 2010

nonn

Log(a(1)) is defined if a(1) > 0, so a(1) = 1.
Log(log(a(2))) is defined if log(a(2)) > 0 => a(2) > 1 => a(2) = 2.
The sequence grows rapidly: a(6) = 2.33150...10^1656520, and is too large to include here.

			a(2) = 2 because log(log(2)) is defined and log(log(1)) is not;
a(3) = 3 because log(log(log(3))) is defined;
a(4) = 16 because log(log(log(log(16)))) is defined.
From _Robert G. Wilson v_, Jul 05 2022: (Start)
a(3) = ceiling(A001113).
a(4) = ceiling(A073226).
a(5) = ceiling(A073227).
a(6) = ceiling(A085667). (End)

Cf. A074785 (log(log(2))), A085667, A004002, A001113, A073226, A073227, A085667.

a(n) = my(k=1); while(1, my(s=k, i=0); while(s > 0, s=log(s); if(s > 0, i++)); if(i==n-1, return(k)); k++) \\ Felix Fröhlich, Nov 22 2015

For n > 2, a(n) = ceiling(e^(e^(...))) where e appears n-2 times.

A329458 Decimal expansion of x such that x^x * log(x) - x^x + 1 = 0, x > 1.

Original entry on oeis.org

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

Offset: 1

Rick Novile, Nov 16 2019

Equivalent to the coordinates of the self-intersection point of the graph y^x - x^y = y - x, where x, y > 1.

			x = 2.41173993056055928114518919802446413261177356034046370153515467138607...

Cf. A073226 (e^e), A194622, A194623, A194556, A194557.

FindRoot[x^x Log[x] - x^x + 1 == 0, {x, 2.40737, 2.41474}, WorkingPrecision -> 1000]

solve(x=2, 3, x^x * log(x) - x^x + 1) \\ Michel Marcus, Nov 16 2019

solve(x=2, 3, x - exp(1-1/x^x)) \\ Michel Marcus, Jul 14 2020

x^x * log(x) - x^x + 1 = 0; x != 1.
y^x - x^y = y - x; y = x; x != 1.
x = exp(1-x^(-x)); x > 1. - Rick Novile, Jul 14 2020

A342476 Numbers m > 1 such that W(m) > W(k) for all 1 < k < m, where W(k) = omega(k)*log(log(k))/log(k).

Original entry on oeis.org

2, 3, 4, 5, 6, 10, 12, 14, 15, 30, 210, 2310, 30030, 510510, 9699690, 223092870

Offset: 1

Amiram Eldar, Mar 13 2021

Includes the primorials prime(k)# = A002110(k) for 1 <= k <= 9.
Since the maximum of the function f(x) = log(log(x))/log(x) occurs at exp(e) = 15.154... (A073226), 15 is the largest term that is not a primorial.
The corresponding record values are -0.528..., 0.085..., 0.235..., 0.295..., 0.650..., 0.724..., 0.732..., 0.735..., 0.735..., 1.079..., 1.254..., 1.321..., 1.357..., 1.371..., 1.381..., 1.384...

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter V, p. 168.

Hans-Joachim Kanold, Über einige Ergebnisse aus der kombinatorischen Zahlentheorie, Abh. Braunschweig, Wiss. Ges., Vol. 36 (1984), pp. 203-229. See eq. 96, pp. 219-220.

Cf. A001221, A002110, A073226.

w[n_] := PrimeNu[n]*Log[Log[n]]/Log[n]; wmax = -1; seq = {}; Do[w1 = w[n]; If[w1 > wmax, wmax = w1; AppendTo[seq, n]], {n, 2, 2500}]; seq

cp's OEIS Frontend

A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(e*d(n))) - e^gamma * log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).

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A331094 Decimal expansion of Pi^Pi - e^e.

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A331189 Decimal expansion of Pi^Pi + e^e.

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A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

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A096181 Floor (e^(n / log(n))).

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A096194 Engel expansion of exp(e).

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A116907 Continued fraction expansion for e^(-e) = 0.0659880358453125370767901875.

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A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

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PARI

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A329458 Decimal expansion of x such that x^x * log(x) - x^x + 1 = 0, x > 1.

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PARI

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A342476 Numbers m > 1 such that W(m) > W(k) for all 1 < k < m, where W(k) = omega(k)*log(log(k))/log(k).

A309968 Numbers n > 1 that give record values for f(n) = sigma(n)/n - e^gamma * log(log(ed(n))) - e^gamma log(log(log(e^e * d(n)))), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203).