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%I #4 Apr 03 2015 12:41:46
%S 3,7,13,21,29,40,52,65,80,97,115,135,157,180,204,230,258,287,318,350,
%T 384,420,457,496,536,578,621,666,713,761,810,862,915,969,1025,1082,
%U 1142,1202,1264,1328,1394,1461,1529,1599,1671,1744,1819,1895,1973
%N a(n) is the n-th sign-change in the sequence of the constants b(n) defined as the coefficients of a Newton interpolation series associated to zeta(s)-1/(s-1).
%H Philippe Flajolet and Linas Vepstas, <a href="http://arxiv.org/abs/math/0611332">On Differences of Zeta Values</a>, arXiv:math/0611332 [math.CA] 2007
%F zeta(s)-1/(s-1) = Sum_{n>=0} (-1)^n*b(n)*binomial(s,n).
%F b(n) = n*(1-EulerGamma - H(n-1)) - 1/2 + Sum_{k=2..n} binomial(n,k)*(-1)^k*zeta(k), where H(n) is the n-th harmonic number.
%e b(1) = 1/2 - EulerGamma < 0,
%e b(2) = -1/2 - 2*EulerGamma + Pi^2/6 < 0,
%e b(3) = -1/2 + 3*(-1/2 - EulerGamma) + Pi^2/2 - zeta(3) > 0, so a(1) = 3.
%t nmax = 2000; $MaxExtraPrecision = 1000; b[n_] := b[n] = n*(1 - EulerGamma - HarmonicNumber[n-1]) - 1/2 + Sum[Binomial[n, k]*(-1)^k*Zeta[k], {k, 2, n}]; Reap[ For[n = 1, n <= nmax, n++, If[b[n] < 0 < b[n+1] || b[n] > 0 > b[n+1], Print[n+1]; Sow[n+1]]]][[2, 1]]
%Y Cf. A001620, A002117.
%K nonn
%O 1,1
%A _Jean-François Alcover_, Apr 03 2015
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