Search: a229172 -id:a229172
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A229168
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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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+10
6
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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
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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.
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MAPLE
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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;
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A229171
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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.
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+10
5
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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
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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LINKS
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EXAMPLE
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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.
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MAPLE
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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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PROG
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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 */
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A229173
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Define a sequence of real numbers by b(1)=e, b(n+1) = b(n) + log(b(n)); a(n) = ceiling( b(n) ).
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+10
4
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3, 4, 6, 7, 9, 11, 14, 16, 19, 22, 25, 28, 31, 35, 38, 42, 46, 49, 53, 57, 61, 65, 70, 74, 78, 83, 87, 91, 96, 100, 105, 110, 114, 119, 124, 129, 134, 139, 143, 148, 153, 158, 163, 169, 174, 179, 184, 189, 194, 200, 205, 210, 216, 221, 226, 232, 237, 243, 248, 254, 259, 265, 270, 276, 282, 287, 293, 299, 304, 310, 316
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graph;
refs;
listen;
history;
text;
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OFFSET
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1,1
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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