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Define a sequence of real numbers by b(1)=2, b(n+1) = b(n) + log_2(b(n)); a(n) = smallest i such that b(i) >= 2^n.
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6
1, 3, 5, 7, 11, 17, 27, 44, 74, 127, 225, 402, 728, 1333, 2459, 4566, 8525, 15993, 30122, 56936, 107953, 205253, 391223, 747369, 1430648, 2743721, 5270959, 10141978, 19542806, 37708232, 72849931, 140905791, 272836175, 528832794, 1026008203, 1992390617
EXAMPLE
The initial terms of the b(n) sequence are approximately
2, 3.00000000000000000000000, 4.58496250072115618145375, 6.78187243514238888864578, 9.54355608312733448665509, 12.7980830210090262451102, 16.4759388461842196480290, 20.5182276175427023220954, 24.8770618274970204646817, 29.5138060245244394221195, 34.3971240984210783617324, ...
b(5) is the first term >= 8, so a(3) = 5.
MAPLE
Digits:=24;
log2:=evalf(log(2));
lis:=[2]; a:=2;
t1:=[1]; l:=2;
for i from 2 to 128 do
a:=evalf(a+log(a)/log2);
if a >= 2^l then
l:=l+1; t1:=[op(t1), i]; fi;
lis:=[op(lis), a];
od:
lis;
map(floor, lis);
map(ceil, lis);
t1;
PROG
(PARI) n=1; p2=2^n; m=2; lg2=log(2); for(i=1, 1992390617, if(m>=p2, print(n " " i); n++; p2=2^n); m=m+log(m)/lg2) /* Donovan Johnson, Oct 04 2013 */
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.
+0
5
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
EXAMPLE
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.
MAPLE
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;
PROG
(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 */
Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = floor( b(n) ).
+0
3
2, 3, 5, 6, 8, 10, 13, 15, 18, 21, 24, 27, 30, 34, 37, 41, 45, 48, 52, 56, 60, 64, 69, 73, 77, 82, 86, 90, 95, 99, 104, 109, 113, 118, 123, 128, 133, 138, 142, 147, 152, 157, 162, 168, 173, 178, 183, 188, 193, 199, 204, 209, 215, 220, 225, 231, 236, 242, 247, 253, 258, 264, 269, 275, 281, 286, 292, 298, 303, 309, 315
Decimal expansion of e + 1 + log(e+1).
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3
5, 0, 3, 1, 5, 4, 3, 5, 1, 5, 9, 7, 7, 2, 6, 8, 0, 6, 9, 4, 0, 9, 2, 8, 2, 9, 6, 6, 3, 2, 0, 5, 1, 8, 1, 3, 9, 6, 7, 2, 5, 2, 7, 1, 7, 9, 3, 7, 0, 3, 0, 7, 9, 4, 9, 6, 8, 6, 0, 3, 1, 1, 4, 2, 5, 6, 1, 2, 7, 1, 5, 3, 1, 8, 6, 2, 7, 3, 4, 8, 1, 5, 5, 0, 7, 5, 8, 2, 5, 8, 4, 8, 0, 2, 4, 5, 8, 8, 1, 8, 8, 1, 6, 4, 8, 7, 4
COMMENTS
This is the third term in the sequence of real numbers discussed in A229171- A229173.
EXAMPLE
5.0315435159772680694092829663205181396725271793703079496860...
MATHEMATICA
RealDigits[E+1+Log[E+1], 10, 120][[1]] (* Harvey P. Dale, Dec 03 2014~ *)
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