A229171 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.
1, 5, 10, 20, 41, 86, 192, 441, 1039, 2493, 6072, 14960, 37199, 93193, 234957, 595562, 1516639, 3877905, 9950908, 25615654, 66127187, 171144672, 443966371, 1154115393, 3005950908
Offset: 1
Examples
The initial terms of the b(n) sequence are approximately 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, ... b(5) is the first term >= e^2, so a(2) = 5.
Programs
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Maple
# A229171, A229172, A229173. Digits:=24; e:=evalf(exp(1)); lis:=[e]; a:=e; t1:=[1]; l:=2; for i from 2 to 128 do a:=evalf(a+log(a)); if a >= e^l then l:=l+1; t1:=[op(t1),i]; fi; lis:=[op(lis),a]; od: lis; map(floor,lis); map(ceil,lis); t1;
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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 */
Extensions
a(7)-a(25) from Donovan Johnson, Oct 04 2013