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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

%I #38 Apr 06 2020 06:10:23

%S 1,2,4,10,26,69,186,504,1369,3720,10111,27483,74705,203068,551995,

%T 1500477,4078718,11087104,30137872,81923228,222690421,605335323,

%U 1645472007,4472856655,12158484965,33050188741,89839727480,244209698681,663830786257,1804479163453,4905082919846

%N 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.

%C 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

%H Jinyuan Wang, <a href="/A081881/b081881.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) is asymptotic to C*exp(n) where C=0.1688... - _Benoit Cloitre_, Apr 14 2003

%F C = 0.16885635666714420373167977550090103410150395689764... (cf. A300897). - _Jon E. Schoenfield_, Apr 12 2018

%F 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

%e 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.

%e 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}, ...

%t res ={}; FoldList[If[ #1+#2 > 1, AppendTo[res, #2];#2, #1+#2]&, 0, Table[1/k, {k, 1, 1000}]]; 1/res

%t 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 *)

%o (PARI) default(realprecision, 10^4); e=exp(1);

%o A136616(k) = floor(e*k + (e-1)/2 + (e-1/e)/(24*k+12));

%o lista(nn) = {my(k=1); print1(k); for(n=2, nn, k=A136616(k-1)+1; print1(", ", k)); } \\ _Jinyuan Wang_, Feb 20 2020

%Y Cf. A002387, A136616, A136617, A295572, A300897.

%K nonn,nice

%O 1,2

%A _Wouter Meeussen_, Apr 13 2003

%E a(13)-a(25) from _Robert G. Wilson v_, Aug 19 2008

%E More terms from _Jinyuan Wang_, Feb 20 2020

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)