

A080281


Numbers k such that Pi^k  1/phi is closer to its nearest integer than any value of Pi^j  1/phi for 1 <= j < k.


1



1, 2, 4, 8, 17, 19, 23, 35, 221, 424, 3846, 16708, 19142, 19937, 55188, 87368
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OFFSET

1,2


COMMENTS

phi is the Golden ratio (1 + sqrt(5))/2.
At k = 3846 the discrepancy is 0.0000887984081945...
At n = 16708 the discrepancy from an integer is 0.00006159...
At n = 19142 the discrepancy from an integer is 0.00003501...
At n = 19937 the discrepancy from an integer is 0.00001498...
At n = 55188 the discrepancy from an integer is 0.00001048...
At n = 87368 the discrepancy from an integer is 0.00000693...
(End)
As 1/phi = phi  1, the sequence is equivalent to "Numbers k such that Pi^k  phi is closer to its nearest integer than any value of Pi^j  phi for 1 <= j < k."  David A. Corneth, Nov 19 2018


LINKS



EXAMPLE

The first term is 1 because this is just Pi  1/phi = 2.52355...
The second term is 2 because Pi^2  1/phi = 9.25157...
The next term is 4 because Pi^4  1/phi is closer to an integer than Pi^3  1/phi.


MATHEMATICA

$MaxExtraPrecision = 10^6; p = 2/(1+Sqrt[5]); b = 1; Do[a = Abs[N[Round[Pi^n  p]  (Pi^n  p), 30]]; If[a < b, Print[n]; b = a], {n, 1, 10^5}] (* Ryan Propper, Jul 27 2005 *)


PROG

(PARI) upto(n) = my(c = 2, phi = (1 + sqrt(5)) / 2, res = List, r = 2); Pik = 1; for(i = 1, n, Pik *= Pi; c = frac(Pik  phi); c = min(c, 1c); if(c < r, listput(res, i); r = c)); res \\ David A. Corneth, Nov 19 2018


CROSSREFS



KEYWORD

more,nonn


AUTHOR

Mark Hudson (mrmarkhudson(AT)hotmail.com), Feb 13 2003


EXTENSIONS



STATUS

approved



