A295572 -id:A295572 - 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

A295571 a(n) = A081881(n) - 1.

Original entry on oeis.org

0, 1, 3, 9, 25, 68, 185, 503, 1368, 3719, 10110, 27482, 74704, 203067, 551994, 1500476, 4078717, 11087103, 30137871, 81923227, 222690420, 605335322, 1645472006, 4472856654, 12158484964, 33050188740, 89839727479, 244209698680, 663830786256, 1804479163452, 4905082919845

Offset: 1

N. J. A. Sloane, Nov 30 2017

nonn

The blocks of fractions described in A081881 extend from 1/A081881(k) through 1/a(k+1) and contain A295572(k) terms. For example the third block is 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/6 and has length 6.

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

Cf. A081881, A295572.

More terms from Jinyuan Wang, Feb 20 2020

A300897 Decimal expansion of lim_{n->infinity} A081881(n)/exp(n).

Original entry on oeis.org

1, 6, 8, 8, 5, 6, 3, 5, 6, 6, 6, 7, 1, 4, 4, 2, 0, 3, 7, 3, 1, 6, 7, 9, 7, 7, 5, 5, 0, 0, 9, 0, 1, 0, 3, 4, 1, 0, 1, 5, 0, 3, 9, 5, 6, 8, 9, 7, 6, 4, 9, 2, 2, 2, 3, 7, 7, 2, 2, 5, 5, 2, 2, 7, 1, 4, 1, 7, 5, 3, 3, 0, 3, 0, 3, 3, 0, 1, 2, 8, 7, 7, 6, 7, 1, 0, 7

Offset: 0

Jon E. Schoenfield, Apr 12 2018

			0.16885635666714420373167977550090103410150395689764...

Cf. A081881, A295572.

A331030 Divide the terms of the harmonic series into groups sequentially so that the sum of each group is minimally greater than 1. a(n) is the number of terms in the n-th group.

Original entry on oeis.org

2, 5, 13, 36, 98, 266, 723, 1965, 5342, 14521, 39472, 107296, 291661, 792817, 2155100, 5858169, 15924154, 43286339, 117664468, 319845186, 869429357, 2363354022, 6424262292, 17462955450, 47469234471, 129034757473, 350752836478, 953445061679, 2591732385596

Offset: 1

Pablo Hueso Merino and Alejandro Argüelles Trujillo, Jan 07 2020

nonn

a(n) = A046171(n+1) through a(5), and grows similarly for n > 5.

Let b(n) = Sum_{j=1..n} a(n); then for n >= 2 it appears that b(n) = round((b(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) - ..., where gamma is the Euler-Mascheroni constant, A001620). Cf. A081881. - Jon E. Schoenfield, Jan 10 2020

			a(1)=2 because 1 + 1/2 = 1.5 > 1,
a(2)=5 because 1/3 + 1/4 + 1/5 + 1/6 + 1/7 = 1.0928... > 1,
etc.

Cf. A002387, A046171, A081881, A096005, A180224, A295572, A331028.

lista(lim=oo)={my(s=0, p=0); for(i=1, lim, s+=1/i; if(s>1, print1(i-p, ", "); s=0; p=i))} \\ Andrew Howroyd, Jan 08 2020

x = 0.0
y = 0.0
z = 0.0
for i in range(1,100000000000000000000000):
  y += 1
  x = x + 1/i
  z = z + 1/i
  if x > 1:
    print(y)
    y = 0
    x = 0

a(1)=2, a(n) = (min(p) : Sum_{s=r..p} 1/s > 1)-r+1, r=Sum_{k=1..n-1} a(k).

a(25)-a(29) from Jon E. Schoenfield, Jan 10 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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A295571 a(n) = A081881(n) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A300897 Decimal expansion of lim_{n->infinity} A081881(n)/exp(n).

Views

Author

Keywords

Examples

Crossrefs

A331030 Divide the terms of the harmonic series into groups sequentially so that the sum of each group is minimally greater than 1. a(n) is the number of terms in the n-th group.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Python

Formula

Extensions