%I #12 Nov 19 2018 07:19:38

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

%N 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.

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

%C At k = 3846 the discrepancy is 0.0000887984081945...

%C From _Ryan Propper_, Jul 27 2005: (Start)

%C At n = 16708 the discrepancy from an integer is 0.00006159...

%C At n = 19142 the discrepancy from an integer is 0.00003501...

%C At n = 19937 the discrepancy from an integer is 0.00001498...

%C At n = 55188 the discrepancy from an integer is 0.00001048...

%C At n = 87368 the discrepancy from an integer is 0.00000693...

%C (End)

%C 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

%e The first term is 1 because this is just Pi - 1/phi = 2.52355...

%e The second term is 2 because Pi^2 - 1/phi = 9.25157...

%e The next term is 4 because Pi^4 - 1/phi is closer to an integer than Pi^3 - 1/phi.

%t $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 *)

%o (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

%Y Cf. A079490, A080052, A080279, A080280.

%K more,nonn

%O 1,2

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

%E a(12)-a(16) from _Ryan Propper_, Jul 27 2005