A256588 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).
3, 7, 13, 21, 29, 40, 52, 65, 80, 97, 115, 135, 157, 180, 204, 230, 258, 287, 318, 350, 384, 420, 457, 496, 536, 578, 621, 666, 713, 761, 810, 862, 915, 969, 1025, 1082, 1142, 1202, 1264, 1328, 1394, 1461, 1529, 1599, 1671, 1744, 1819, 1895, 1973
Offset: 1
Keywords
Examples
b(1) = 1/2 - EulerGamma < 0, b(2) = -1/2 - 2*EulerGamma + Pi^2/6 < 0, b(3) = -1/2 + 3*(-1/2 - EulerGamma) + Pi^2/2 - zeta(3) > 0, so a(1) = 3.
Links
- Philippe Flajolet and Linas Vepstas, On Differences of Zeta Values, arXiv:math/0611332 [math.CA] 2007
Programs
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Mathematica
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]]
Formula
zeta(s)-1/(s-1) = Sum_{n>=0} (-1)^n*b(n)*binomial(s,n).
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.