A059923 a(n+1) is smallest number with a(n+1)^n > a(n)^(n+1).

1, 2, 3, 5, 8, 13, 20, 31, 48, 74, 114, 176, 271, 417, 642, 988, 1521, 2341, 3603, 5545, 8533, 13131, 20207, 31096, 47853, 73639, 113320, 174383, 268350, 412951, 635471, 977896, 1504837, 2315721, 3563551, 5483776, 8438716, 12985930, 19983416

Offset

1,2

Comments

A kind of discrete exponential, since the series of n-th differences resembles the original sequence. The principle of construction is F(a(n+1)) > G(a(n)) as in A059842 but slightly modified to F(n,a(n+1)) > F(n+1,a(n)) with F(n,x) = x^n. This seems to be a fruitful construction principle!

Links

T. D. Noe, Table of n, a(n) for n = 1..300

Formula

a(n+1) = floor( a(n)^(1+1/n) ) + 1; compare A080870. - Paul D. Hanna, Feb 21 2003

a(n) = floor(x^n), where x=1.53885131519173... - Paul D. Hanna, Feb 21 2003

Example

We have a(5)=8 and therefore a(6) = 13 because 13^5 > 8^6.

Maple

a := proc(n) option remember: if n=1 then RETURN(1) fi: if n=2 then RETURN(2) fi: ceil(a(n-1)^((n)/(n-1))): end: Digits := 20: for n from 1 to 250 do printf(`%d, `, a(n)) od:

Mathematica

a[n_] := a[n] = Floor[a[n-1]^(1+1/(n-1))]+1; a[1] = 1; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 03 2014, after Paul D. Hanna *)
nxt[{n_, a_}]:={n+1, Floor[a^(1+1/n)]+1}; NestList[nxt, {1, 1}, 40][[All, 2]] (* Harvey P. Dale, Aug 11 2019 *)

Prog

(PARI) { default(realprecision, 200); a=1; for (n=1, 300, write("b059923.txt", n, " ", a); a=floor(a^(1 + 1/n)) + 1; ) } \\ Harry J. Smith, Jun 30 2009

Crossrefs

Cf. A059842, A080869, A080870.

Sequence in context: A216053 A361721 A173597 * A266331 A055805 A023437

Adjacent sequences: A059920 A059921 A059922 * A059924 A059925 A059926

Keyword

easy,nice,nonn

Author

Rainer Rosenthal, Mar 03 2001

Extensions

More terms from James A. Sellers, Mar 15 2001

Status

approved