|
| |
|
|
A081881
|
|
Pack bins of size 1 sequentially with items of size 1/1, 1/2, 1/3, 1/4, ... . Sequence gives values of n for which 1/n starts a new bin.
|
|
7
|
|
|
|
1, 2, 4, 10, 26, 69, 186, 504, 1369, 3720, 10111, 27483, 74705, 203068, 551995, 1500477, 4078718, 11087104, 30137872, 81923228, 222690421, 605335323, 1645472007, 4472856655, 12158484965, 33050188741, 89839727480, 244209698681, 663830786257, 1804479163453, 4905082919846
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
For n >= 3, it appears that a(n) = round((a(n-1) - 1/2)*e). Verified through n = 10000 (using the approximation Sum_{j=1..k} 1/j = log(k) + gamma + 1/(2*k) - 1/(12*k^2) + 1/(120*k^4) - 1/(252*k^6) + 1/(240*k^8) - ... + 7709321041217/(16320*k^32), where gamma is the Euler-Mascheroni constant, A001620). - Jon E. Schoenfield, Mar 30 2018
|
|
|
LINKS
|
|
|
|
FORMULA
|
a(n) is asymptotic to C*exp(n) where C=0.1688... - Benoit Cloitre, Apr 14 2003
a(n) = 1 + (A136616^(n-1))(0), where (f^0)(x)=x, (f^(n+1))(x) = f((f^n)(x)) for any function f. - Rainer Rosenthal, Feb 16 2008, Apr 05 2020
|
|
|
EXAMPLE
|
1/1; 1/2+1/3, 1/4+1/5+1/6+1/7+1/8+1/9 are all just less than or equal to 1; so first four terms are 1, 2, 4, 10.
Lower and upper indices of bin contents are {1,1}, {2,3}, {4,9}, {10,25}, {26,68}, {69,185}, {186,503}, {504,1368}, {1369,3719}, {3720,10110}, {10111,27482}, ...
|
|
|
MATHEMATICA
|
res ={}; FoldList[If[ #1+#2 > 1, AppendTo[res, #2]; #2, #1+#2]&, 0, Table[1/k, {k, 1, 1000}]]; 1/res
lst = {1, 2}; n = 2; Do[s = 0; While[s = N[s + 1/n, 64]; s < 1, n++ ]; AppendTo[lst, n]; Print@n, {i, 25}]; lst (* Robert G. Wilson v, Aug 19 2008 *)
|
|
|
PROG
|
(PARI) default(realprecision, 10^4); e=exp(1);
A136616(k) = floor(e*k + (e-1)/2 + (e-1/e)/(24*k+12));
lista(nn) = {my(k=1); print1(k); for(n=2, nn, k=A136616(k-1)+1; print1(", ", k)); } \\ Jinyuan Wang, Feb 20 2020
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,nice
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
