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 1-3 of 3 results.

A088346 Smallest integer k for which exp(x) > x^n, for all x>=k, n>=3.

Original entry on oeis.org

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
Offset: 3

Views

Author

Roger L. Bagula, Nov 07 2003

Keywords

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).

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)

Programs

  • 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]

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

A190050 Expansion of ((1-x)*(3*x^2-3*x+1))/(1-2*x)^3.

Original entry on oeis.org

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
Offset: 0

Views

Author

Johannes W. Meijer, May 06 2011

Keywords

Comments

The second left hand column of triangle A175136.

Links

Crossrefs

Cf. A175136, A011782, A190951.
Related to A001788.

Programs

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

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.

A190952 Largest integer k for which exp(k) < k^n, n>=3.

Original entry on oeis.org

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
Offset: 3

Views

Author

Shel Kaphan, May 24 2011

Keywords

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

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)

Programs

  • Mathematica
    a[n_] := Floor[E^-ProductLog[-1, -1/n]]; Table[a[n], {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
Showing 1-3 of 3 results.

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