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

0,2

Comments

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

References

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

Links

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

Formula

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

Mathematica

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

Crossrefs

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

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

Sequence in context: A299508 A299308 A300109 * A295758 A381142 A343707

Adjacent sequences: A056050 A056051 A056052 * A056054 A056055 A056056

Keyword

nonn

Author

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

Extensions

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

Status

approved