A190951 -id:A190951

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A088346

Smallest integer k for which exp(x) > x^n, for all x>=k, n>=3.

+10
5

5, 9, 13, 17, 22, 27, 31, 36, 41, 46, 52, 57, 62, 68, 73, 79, 85, 90, 96, 102, 108, 114, 120, 126, 132, 138, 145, 151, 157, 164, 170, 176, 183, 189, 196, 202, 209, 215, 222, 229, 235, 242, 249, 255, 262, 269, 276, 283, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

n=3 is the starting index since exp(x) > x^n for all x>=0 when n=1,2.

This function also cancels out a different set of numbers from the factorial than the primes using the asymptotic behavior of prime(n) and pi(n).

LINKS

Table of n, a(n) for n=3..60.

MATHEMATICA

a[n_] := Ceiling[E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]

(* Also, the following code is from another definition of the *)

(* same sequence. *)

(* asymptotic prime like product function*) p[n_]=n!/(2*Product[Floor[i*Log[i]], {i, 2, Floor[n/Log[n]]}])

a0=Table[Floor[p[n]/p[n-1]], {n, 3, 500}];

(* composite like distribution*) Delete[Union[a0], 1];

(* pick of prime like numbers *) c=Table[If[a0[[n]]==1, n+2, 0], {n, 1, digits-3}];

d=Delete[Union[c], 1]

CROSSREFS

Cf. A190951 (Closest integer to the largest real x such that exp(x) = x^n)

Cf. A190952 (Largest integer k for which exp(k) < k^n)

KEYWORD

nonn

AUTHOR

Roger L. Bagula, Nov 07 2003

EXTENSIONS

Partially edited Charles R Greathouse IV, Nov 02 2009

Provided new name, and added 2 initial terms, by Shel Kaphan, May 20 2011

Added Mathematica function, by Shel Kaphan, May 23 2011

Reverted to starting at n=3, improved Mathematica code, by Shel Kaphan, May 24 2011

STATUS

approved

A190050

Expansion of ((1-x)*(3*x^2-3*x+1))/(1-2*x)^3

+10
3

1, 2, 6, 17, 46, 120, 304, 752, 1824, 4352, 10240, 23808, 54784, 124928, 282624, 634880, 1417216, 3145728, 6946816, 15269888, 33423360, 72876032, 158334976, 342884352, 740294656, 1593835520, 3422552064

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The second left hand column of triangle A175136.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-12,8).

FORMULA

G.f.: ((1-x)*(3*x^2-3*x+1))/(1-2*x)^3.

a(n) = (n^2 + 5*n + 10)*2^(n-4) for n >=1 with a(0)=1.

a(n) = A001788(n+1) -4*A001788(n) +6*A001788(n-1) -3*A001788(n-2) for n >=1 with a(0)=1.

MAPLE

A190050:= proc(n) option remember; if n=0 then A190050(n):=1: else A190050(n):=(n^2+5*n+10)*2^(n-4) fi: end: seq (A190050(n), n=0..26);

MATHEMATICA

Join[{1}, LinearRecurrence[{6, -12, 8}, {2, 6, 17}, 30]] (* or *) CoefficientList[Series[((1-x)*(3*x^2-3*x+1))/(1-2*x)^3, {x, 0, 50}], x] (* G. C. Greubel, Jan 10 2018 *)

PROG

(PARI) x='x+O('x^30); Vec(((1-x)*(3*x^2-3*x+1))/(1-2*x)^3) \\ G. C. Greubel, Jan 10 2018

(PARI) for(n=0, 30, print1(if(n==0, 1, (n^2 + 5*n + 10)*2^(n-4)), ", ")) \\ G. C. Greubel, Jan 10 2018

(Magma) [1] cat [(n^2 + 5*n + 10)*2^(n-4): n in [1..30]]; // G. C. Greubel, Jan 10 2018

CROSSREFS

Cf. A175136, A011782, A190951.

Related to A001788.

KEYWORD

nonn,easy

AUTHOR

Johannes W. Meijer, May 06 2011

STATUS

approved

A190952

Largest integer k for which exp(k) < k^n, n>=3.

+10
2

4, 8, 12, 16, 21, 26, 30, 35, 40, 45, 51, 56, 61, 67, 72, 78, 84, 89, 95, 101, 107, 113, 119, 125, 131, 137, 144, 150, 156, 163, 169, 175, 182, 188, 195, 201, 208, 214, 221, 228, 234, 241, 248, 254, 261, 268, 275, 282, 288, 295, 302, 309, 316, 323, 330, 337, 344, 351

(list; graph; refs; listen; history; text; internal format)

OFFSET

3,1

COMMENTS

n=3 is the starting index because exp(x)>x^n for all x>=0 when n=1,2.

Conjecture: There are floor((n+1)/log(n+1))-2 terms less than or equal to n. - Benedict W. J. Irwin, Jun 15 2016

LINKS

Table of n, a(n) for n=3..60.

FORMULA

Conjecture: G.f.: Sum_{ j>=1 } (Sum_{ k>=1 } x^(j+floor((k+1)/log(k+1)))) + x^j. - Benedict W. J. Irwin, Jun 15 2016

a(n) = floor(-n*LambertW(-1,-1/n)). - Vaclav Kotesovec, Jun 29 2016

MATHEMATICA

a[n_] := Floor[E^-ProductLog[-1, -1/n]]; Table[a[n], {n, 3, 60}]

CROSSREFS

Cf. A088346 (Smallest integer k where exp(x)>x^n for all x>=k)

Cf. A190951 (Closest integer to the largest real x such that exp(x) = x^n)

KEYWORD

nonn

AUTHOR

Shel Kaphan, May 24 2011

STATUS

approved

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