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%I #39 Dec 10 2022 18:34:50
%S 3,15,113,837,6183,45691,337607,2494595,18432707,136200301,1006391657,
%T 7436284415,54947122715,406007372211,3000011249847,22167251422541,
%U 163795064320249,1210290918990281,8942907496445513,66079645178783351
%N 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.
%C 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
%D Calvin C. Clawson, "Mathematical Mysteries, The Beauty and Magic of Numbers," Plenum Press, NY and London, 1996, page 64.
%H Grant Sanderson, <a href="https://www.youtube.com/watch?v=851U557j6HE">Researchers thought this was a bug (Borwein integrals)</a>, 3Blue1Brown video (2022).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Borwein_integral">Borwein integral</a>
%F a(n) ~ floor((1/2)*A002387(2n)).
%F The next term is approximately the previous term * e^2.
%F 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
%F a(n) ~ exp(2*n - A350763) = (1/2)*exp(2*n - gamma), gamma = A001620. - _A.H.M. Smeets_, Apr 15 2022
%t s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 11}]
%Y Cf. A002387, A056054, A091463, A091464, A091465.
%Y Cf. also A092318, A092317, A092315, A281355, A074599, A025547.
%K nonn
%O 1,1
%A _Robert G. Wilson v_, Jul 25 2000 and Jan 11 2004
%E Corrected by _N. J. A. Sloane_, Feb 16, 2004
%E More terms from _Robert G. Wilson v_, Apr 17 2004
%E a(17) corrected - see correction in A092315. - _Gerhard Kirchner_, Jul 25 2020
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