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 1-5 of 5 results.

A190951 Closest integer to the largest real number x such that exp(x) = x^n, for n>=3.

Original entry on oeis.org

5, 9, 13, 17, 21, 26, 31, 36, 41, 46, 51, 56, 62, 67, 73, 79, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 157, 163, 169, 176, 182, 189, 195, 202, 208, 215, 221, 228, 235, 241, 248, 255, 262, 268, 275, 282, 289, 296, 303, 310, 317, 324, 331, 338, 345, 352
Offset: 3

Views

Author

Shel Kaphan, May 23 2011

Keywords

Comments

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

Crossrefs

Cf. A088346 (Smallest integer k where exp(x)>x^n for all x>=k).
Cf. A190952 (Largest integer k for which exp(k) < k^n).

Programs

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

A262058 Least integer k>1 such that sqrt(k)/log(k) exceeds n.

Original entry on oeis.org

2, 2, 289, 681, 1280, 2109, 3190, 4538, 6170, 8100, 10339, 12901, 15795, 19032, 22620, 26570, 30888, 35583, 40662, 46133, 52003, 58277, 64962, 72065, 79590, 87544, 95932, 104759, 114030, 123750, 133924, 144557, 155652, 167215, 179250, 191760
Offset: 1

Views

Author

Robert G. Wilson v, Sep 09 2015

Keywords

Links

Crossrefs

Cf. A088346, A262059, A262060.

Programs

  • Maple
    A262058 := proc(n)
    Digits := 30 ;
    for k from 2 do
        if evalf(sqrt(k) > n*log(k)) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 22 2015
  • Mathematica
    f[n_] := f[n] = Block[{k = f[n - 1]}, While[n > Sqrt[k]/Log[k], k++]; k]; f[1] = 2; Array[f, 50]
  • PARI
    a(n) = {my(k = 2); while(sqrt(k)/log(k) <= n, k++); k;} \\ Michel Marcus, Sep 10 2015
  • Sage
    def A262058(n,d=50):
    (low,high) = (1,2)
    while N(sqrt(high),digits=d) <= N(n*log(high),digits=d):
        high *= 2
    while low+1Danny Rorabaugh, Sep 26 2015

A262059 Least integer k such that k^(1/3)/log(k) exceeds n.

Original entry on oeis.org

2, 4913, 29410, 96854, 236916, 484596, 879483, 1465239, 2289183, 3401984, 4857388, 6712006, 9025131, 11858570, 15276512, 19345406, 24133846, 29712478, 36153913, 43532644, 51924974, 61408954, 72064316, 83972419, 97216198, 111880113, 128050105, 145813554, 165259239, 186477301, 209559205
Offset: 1

Views

Author

Robert G. Wilson v, Sep 09 2015

Keywords

Crossrefs

Cf. A088346, A262058, A262060.

Programs

  • Maple
    A262059val := proc(k)
    Digits := 100 ;
    evalf(root[3](k)/log(k)) ;
end proc:
A262059lims := proc(n,lowk,highk)
    local vallow, valhigh,midk,valmid ;
    vallow := A262059val(lowk) ;
    valhigh := A262059val(highk) ;
    if valhigh > n and vallow <= n and highk= lowk+1 then
        return highk;
    else
        midk := floor((lowk+highk)/2) ;
        valmid := A262059val(midk) ;
        if valmid < n then
            return procname(n,midk,highk) ;
        else
            return procname(n,lowk,midk) ;
        end if;
    end if;
end proc:
A262059 := proc(n)
    local lowk,highk,p ;
    if n = 1 then
        return 2;
    end if;
    for p from 0 do
        lowk := 10^p ;
        highk := 10^(p+1) ;
        if A262059val(highk) >=n then
            return A262059lims(n,min(2,lowk),highk) ;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 22 2015
  • Mathematica
    f[n_] := f[n] = Block[{k = f[n - 1]}, While[n > k^(1/3)/Log[k], k++]; k]; f[1] = 2; Array[f, 40]
  • PARI
    a(n) = {my(k = 2); while(sqrtn(k,3)/log(k) <= n, k++); k;} \\ Michel Marcus, Sep 10 2015

A262060 Least integer k such that k^(1/n)/log(k) exceeds 2.

Original entry on oeis.org

2, 2, 4913, 463584, 58571590, 9380523077, 1831736082750, 423908600424675, 113798703080610442, 34848887401383308294, 12011778862556061365985, 4609276407921507486293833, 1951202873990586514532224545, 904205931392036935959059378623
Offset: 1

Views

Author

Robert G. Wilson v, Sep 09 2015

Keywords

Links

Crossrefs

Cf. A084238, A088346, A262058, A262059.

Programs

  • Mathematica
    f[n_] := f[n] = Block[{k = f[n - 1]}, While[2 > k^(1/n)/Log[k], k++]; k]; f[1] = 2; Array[f, 6]
  • PARI
    a(n) = {my(k = 2); while(sqrtn(k,n)/log(k) <= 2, k++); k;} \\ Michel Marcus, Sep 10 2015

Extensions

a(14) from Jon E. Schoenfield, Sep 12 2015

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 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.