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, 2, 4, 8, 17, 19, 23, 35, 221, 424, 3846, 16708, 19142, 19937, 55188, 87368
Offset: 1
Examples
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.
Programs
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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 *) -
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, 1-c); if(c < r, listput(res, i); r = c)); res \\ David A. Corneth, Nov 19 2018
Extensions
a(12)-a(16) from Ryan Propper, Jul 27 2005
Comments