A229171 - OEIS

%I #18 Nov 05 2013 20:13:37

%S 1,5,10,20,41,86,192,441,1039,2493,6072,14960,37199,93193,234957,

%T 595562,1516639,3877905,9950908,25615654,66127187,171144672,443966371,

%U 1154115393,3005950908

%N Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = smallest i such that b(i) >= e^n.

%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%e The initial terms of the b(n) sequence are approximately

%e 2.71828182845904523536029, 3.71828182845904523536029, 5.03154351597726806940929, 6.64727031503970856301384, 8.54147660649653209023621, 10.6864105040926911986276, 13.0553833920216929230460, 15.6245839611886549261305, 18.3734295299727029212384, 21.2843351036624388705641, 24.3423064646657059114213, 27.5345223079930416816192, 30.8499628820185220765989, ...

%e b(5) is the first term >= e^2, so a(2) = 5.

%p # A229171, A229172, A229173.

%p Digits:=24;

%p e:=evalf(exp(1));

%p lis:=[e]; a:=e;

%p t1:=[1]; l:=2;

%p for i from 2 to 128 do

%p a:=evalf(a+log(a));

%p if a >= e^l then

%p l:=l+1; t1:=[op(t1),i]; fi;

%p lis:=[op(lis),a];

%p od:

%p lis;

%p map(floor,lis);

%p map(ceil,lis);

%p t1;

%o (PARI) n=1; m=exp(1); mn=m^n; for(i=1, 3005950908, if(m>=mn, print(n " " i); n++; mn=exp(1)^n); m=m+log(m)) /* _Donovan Johnson_, Oct 04 2013 */

%Y Cf. A229172, A229173, A229175; A229168, A229169, A229170, A010062, A229167; also A004207.

%K nonn,more

%O 1,2

%A _N. J. A. Sloane_, Sep 27 2013

%E a(7)-a(25) from _Donovan Johnson_, Oct 04 2013