A091464 -id:A091464 - cp's OEIS frontend

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

1, 3, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351, 488266205223462461, 3607826381608149807

Offset: 0

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

a(2) = 15 and a(3) = 113 are related to the Borwein integrals. Concretely, a(2) = 15 is the smallest odd m such that the integral Integral_{x=-oo..oo} Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi, and a(3) = 113 is the smallest odd m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Calvin C. Clawson, "Mathematical Mysteries, The Beauty and Magic of Numbers," Plenum Press, NY and London, 1996, page 64.

Robert G. Wilson v, Table of n, a(n) for n = 0..500
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Cf. A002387, A056054, A091463, A091464, A091465.

Cf. also A092318, A092317, A092315, A281355, A074599, A025547.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 11}]

a(n) ~ floor((1/2)*A002387(2n)).
The next term is approximately the previous term * e^2.
a(n) = A092315(n)*2 + 1 = floor(exp(n*2-Euler)/4+1/8)*2+1 for all n (conjectured). - M. F. Hasler, Jan 24 2017
a(n) ~ exp(2*n - A350763) = (1/2)*exp(2*n - gamma), gamma = A001620. - A.H.M. Smeets, Apr 15 2022

Corrected by N. J. A. Sloane, Feb 16 2004
More terms from Robert G. Wilson v, Apr 17 2004
a(17) corrected - see correction in A092315. - Gerhard Kirchner, Jul 25 2020
a(0) prepended by Robert G. Wilson v, Oct 23 2024

A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

Original entry on oeis.org

8, 62, 454, 3348, 24734, 182760, 1350428, 9978382, 73730824, 544801200, 4025566630, 29745137662, 219788490858, 1624029488844, 12000044999386, 88669005690160, 655180257281000, 4841163675961122, 35771629985782052

Offset: 1

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

Numbers 2*m such that floor(f(m)) = floor(f(m-1)) where f(m) = Sum_{j=1..m} ((2*j-1)/(2*j)). Examples:
  floor(f(1))=floor(1/2)=0;
  floor(f(2))=floor(1/2+3/4)=floor(1.25)=1, then 2*2=4 is not in the sequence;
  floor(f(3))=floor(1/2+3/4+5/6)=floor(2.083..)=2, then 2*3=6 is not in the sequence;
  floor(f(4))=floor(1/2+3/4+5/6+7/8)=floor(2.958..)=2, then 2*4=8 is the first term of the sequence. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Aug 15 2007

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Plenum Press, NY and London, 1996, page 64.

Cf. A002387, A056053, A091463, A091464, A091465.

Cf. A056054.

s = 0; k = 2; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 12}]
(* or assuming that the Mathematica coding in A002387 is correct then *)
b[n_] := Module[{k = Floor[2a[2n]]}, If[ EvenQ[k], k, k + 1]]; Table[ b[n], {n, 19}] (* Robert G. Wilson v, Apr 17 2004 *)

a(n) = 2*A002387(2n).

The next term is approximately the previous term * e^2.

A091463 a(n) is the smallest j such that 1/1 + 1/4 + 1/7 + ... + 1/j exceeds n.

Original entry on oeis.org

4, 52, 1060, 21301, 427873, 8594032, 172615738, 3467079760, 69638158519

Offset: 1

Robert G. Wilson v, Jan 12 2004

Cf. A002387, A056053, A056054, A091462, A091464.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 7}]

The next term is approximately the previous term * e^3.

Name edited by Jon E. Schoenfield, Dec 20 2019

a(8)-a(9) from Hugo Pfoertner, Dec 27 2019

A091462 a(n) is the smallest j such that 1/3 + 1/6 + 1/9 + ... + 1/j exceeds n.

Original entry on oeis.org

3, 33, 681, 13650, 274140, 5506263, 110596236, 2221384803, 44617706493, 896170591203, 18000067499079, 361541020372644, 7261745513941683, 145856057647068072, 2929597231340774769, 58842533360163495285, 1181883876459465987195, 23738772239546776075803, 476805986328559173414774

Offset: 0

Robert G. Wilson v, Jan 12 2004

nonn

Cf. A002387, A056053, A056054, A091463, A091464.

s = 0; k = 3; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 3]; Print[k]; k += 3, {n, 1, 12}]

a(n) = 3*A002387(3n).

The next term is approximately the previous term * e^3.

a(0) prepended and more terms added by Max Alekseyev, Sep 01 2023

cp's OEIS Frontend

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

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A056054 a(n) = smallest even number 2m such that value of odd harmonic series Sum_{j=0..m} 1/(2j) is > n.

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A091463 a(n) is the smallest j such that 1/1 + 1/4 + 1/7 + ... + 1/j exceeds n.

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Author

Keywords

Crossrefs

Programs

Mathematica

Formula

Extensions

A091462 a(n) is the smallest j such that 1/3 + 1/6 + 1/9 + ... + 1/j exceeds n.

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Author

Keywords

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Programs

Mathematica

Formula

Extensions