A300897 -id:A300897 - cp's OEIS frontend

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.

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

Offset: 1

Wouter Meeussen, Apr 13 2003

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

			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}, ...

Jinyuan Wang, Table of n, a(n) for n = 1..1000

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

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 *)

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

a(n) is asymptotic to C*exp(n) where C=0.1688... - Benoit Cloitre, Apr 14 2003
C = 0.16885635666714420373167977550090103410150395689764... (cf. A300897). - Jon E. Schoenfield, Apr 12 2018
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

a(13)-a(25) from Robert G. Wilson v, Aug 19 2008

More terms from Jinyuan Wang, Feb 20 2020

A295572 First differences of A081881.

Original entry on oeis.org

1, 2, 6, 16, 43, 117, 318, 865, 2351, 6391, 17372, 47222, 128363, 348927, 948482, 2578241, 7008386, 19050768, 51785356, 140767193, 382644902, 1040136684, 2827384648, 7685628310, 20891703776, 56789538739, 154369971201, 419621087576, 1140648377196, 3100603756393

Offset: 1

N. J. A. Sloane, Nov 30 2017, following a suggestion from Loren Booda

nonn

See A081881 and A295571 for discussion.

If the harmonic series is divided into the longest possible consecutive groups so that the sum of each group is <= 1, then a(n) is the number of terms in the n-th group. - Pablo Hueso Merino, Feb 16 2020

			From _Pablo Hueso Merino_, Feb 16 2020: (Start)
a(1) = 1 because 1 <= 1, 1 is one term (if you added 1/2 the sum would be greater than 1).
a(2) = 2 because 1/2 + 1/3 = 0.8333... <= 1, 1/2 and 1/3 are two terms (if you added 1/4 the sum would be greater than one).
a(3) = 6 because 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 = 0.9956... <= 1, it is a sum of six terms. (End)

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Cf. A081881, A295571, A300897, A331028, A331030.

a[1]=1;
a[n_]:= a[n]= Module[{sum = 0}, r = 1 + Sum[a[k], {k, n-1}];
   x = r;
   While[sum <= 1, sum += 1/x++];
   p = x-2;
   p -r +1];
Table[a[n], {n, 10}] (* Pablo Hueso Merino, Feb 16 2020 *)

a(1) = 1, a(n) = (max(m) : Sum_{s=r..m} 1/s <= 1)-r+1, r = Sum_{k=1..n-1} a(k). - Pablo Hueso Merino, Feb 16 2020

a(n) ~ c * exp(n), where c = (exp(1)-1) * A300897 = 0.290142809280953235916025... - Vaclav Kotesovec, Apr 05 2020

More terms from Jinyuan Wang, Feb 20 2020

cp's OEIS Frontend

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.

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