A091560 Fractional part of e^a(n) is the largest yet.

1, 8, 19, 76, 166, 178, 209, 1907, 20926, 22925, 32653, 119136

Offset

1,2

Comments

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of e^m is greater than the fractional part of e^k for all k, 1<=k<m.

The next such number must be greater than 100000. [Hieronymus Fischer, Jan 06 2009]

a(13) > 300,000. Robert Price, Mar 23 2019

Links

Table of n, a(n) for n=1..12.

Formula

Recursion: a(1):=1, a(k):=min{ m>1 | fract(e^m) > fract(e^a(k-1))}, where fract(x) = x-floor(x). [Hieronymus Fischer, Jan 06 2009]

Example

a(2)=8, since fract(e^8)= 0.9579870417..., but fract(e^k)<=0.7182818... for 1<=k<=7;
thus fract(e^8)>fract(e^k) for 1<=k<8 and 8 is the minimal exponent > 1 with this property. [Hieronymus Fischer, Jan 06 2009]

Mathematica

a = 0; Do[b = N[ FractionalPart[ N[ E, 2^12]^n], 24]; If[b > a, Print[n]; a = b], {n, 1, 9400}] (* Robert G. Wilson v, Mar 16 2004 *)

Prog

(PARI) E=exp(1); /* use sufficient precision! */
ym=0; for(i=1, 1000, x=E^i; y=x-floor(x); if(y>ym, print1(", "i); ym=y))

Crossrefs

Cf. A153663, A153671, A153679, A153687, A153695, A153707, A154130, A153711, A153719, A000149. [Hieronymus Fischer, Jan 06 2009]

Sequence in context: A297302 A347125 A057452 * A061877 A297459 A297696

Adjacent sequences: A091557 A091558 A091559 * A091561 A091562 A091563

Keyword

nonn,more

Author

Jon Perry, Mar 04 2004

Extensions

a(8) from Robert G. Wilson v, Mar 16 2004

a(9)-a(11) from Hieronymus Fischer, Jan 06 2009

a(12) from Robert Price, Mar 23 2019

Status

approved