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

Author

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

Keywords

• more
• nonn

Comments

phi is the Golden ratio (1 + sqrt(5))/2.

At k = 3846 the discrepancy is 0.0000887984081945...

From Ryan Propper, Jul 27 2005: (Start)

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

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.
		

Crossrefs

Cf. A079490, A080052, A080279, A080280.

Programs

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