

A002505


Nearest integer to the nth Gram point.
(Formerly M5052 N2185)


8



18, 23, 28, 32, 35, 39, 42, 46, 49, 52, 55, 58, 60, 63, 66, 68, 71, 74, 76, 79, 81, 84, 86, 88, 91, 93, 95, 98, 100, 102, 104, 107, 109, 111, 113, 115, 118, 120, 122, 124, 126
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OFFSET

0,1


COMMENTS

Every integer greater than 3295 is in this sequence.  T. D. Noe, Aug 03 2007
Nearest integer to points t such that Re(zeta(1/2+i*t)) is not equal to zero and Im(zeta(1/2+i*t))=0.  Mats Granvik, May 14 2016


REFERENCES

C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.
A. Ivić, The Theory of Hardy's ZFunction, Cambridge University Press, 2013, pages 109112.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 0..3000
Guilherme França and André LeClair, A theory for the zeros of Riemann Zeta and other Lfunctions, arXiv:1407.4358 [math.NT], 2014, formula (163) at page 47.
Mats Granvik Gram points computed by iterative formula
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function, annotated scanned copy of page 58.
Eric Weisstein's World of Mathematics, Gram Point


FORMULA

a(n) ~ 2*Pi*n/log n.  Charles R Greathouse IV, Oct 23 2015
From Mats Granvik, May 16 2016: (Start)
a(n) = round(2*Pi*exp(1 + LambertW((8*n + 1)/(8*exp(1))))), Eric Weisstein's World of Mathematics.
a(n+1) = round(2*Pi*(n  7/8)/LambertW((n  7/8)/exp(1))), after Guilherme França, André LeClair formula (163) page 47.
(End)
For c = 0 the nth Gram point x is the fixed point solution to the iterative formula:
x = 2*Pi*e^(LambertW (((c  n + RiemannSiegelTheta (x)/Pi + (x*(log (x) + 1 + log (2) + log (Pi)))/(2*Pi) + 2)/e)) + 1).  Mats Granvik, Jun 17 2017


MATHEMATICA

a[n_] := Round[ g /. FindRoot[ RiemannSiegelTheta[g] == Pi*n, {g, 2*Pi*Exp[1 + ProductLog[(8*n + 1)/(8*E)]]}]]; Table[a[n], {n, 0, 40}] (* JeanFrançois Alcover, Oct 17 2012, after Eric W. Weisstein *)


PROG

(Sage)
a = lambda n: round(2*pi*(n  7/8)/lambert_w((n  7/8)/exp(1)))
print [a(n) for n in (1..41)] # Peter Luschny, May 19 2016


CROSSREFS

Cf. A273061. A114857 = 17.8455995..., A114858 = 23.1702827...
Sequence in context: A290172 A031407 A267826 * A182438 A050772 A086473
Adjacent sequences: A002502 A002503 A002504 * A002506 A002507 A002508


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



