A306972 Nonsquare numbers n such that the difference between sqrt(n) and its best rational approximation as x/y with y<=n produces a new minimum of abs(sqrt(n)-x/y). x/y is provided as A306973/A306974.

2, 3, 5, 8, 10, 15, 17, 19, 20, 21, 28, 44, 68, 72, 90, 107, 114, 115, 156, 159, 164, 267, 278, 294, 336, 427, 455, 464, 592, 644, 665, 754, 998, 1040, 1139, 1173, 1278, 1314, 1335, 1463, 1503, 1593, 1943, 2175, 2262, 2520, 2642, 2873, 2978, 3003, 3047, 3181, 3197

Offset

1,1

Links

Table of n, a(n) for n=1..53.

Example

   k s=sqrt(k)     x/y  |s - x/y|
   2 1.4142135...  3/2  0.0857864... new minimum
   3 1.7320508...  5/3  0.0653841... new minimum
   5 2.2360679...  9/4  0.0139320... new minimum
   6 2.4494897...  5/2  0.0505102...
   7 2.6457513...  8/3  0.0209153...
   8 2.8284271... 17/6  0.0049062... new minimum
  10 3.1622776... 19/6  0.0043890... new minimum
  11 3.3166247... 10/3  0.0167085...
  12 3.4641016...  7/2  0.0358983...
  13 3.6055512... 18/5  0.0055512...
  14 3.7416573... 15/4  0.0083426...
  15 3.8729833... 31/8  0.0020166... new minimum
  17 4.1231056... 33/8  0.0018943... new minimum
  18 4.2426406... 17/4  0.0073593...
  19 4.3588989... 61/14 0.0017560... new minimum
  20 4.4721359... 76/17 0.0015477... new minimum
.
  a(1)..a(9)    = [2, 3, 5,  8, 10, 15, 17, 19, 20]
  A306973(1..9) = [3, 5, 9, 17, 19, 31, 33, 61, 76]
  A306974(1..9) = [2, 3, 4,  6,  6,  8,  8, 14, 17]

Prog

(PARI) dmin=1; for(k=2, 3200, if(!issquare(k), s=sqrt(k); d=abs(s-bestappr(s, k)); if(d<dmin, dmin=d; print1(k, ", "))))

Crossrefs

Cf. A306973, A306974.

Sequence in context: A022955 A363076 A254297 * A087279 A246346 A084907

Adjacent sequences: A306969 A306970 A306971 * A306973 A306974 A306975

Keyword

nonn

Author

Hugo Pfoertner, Mar 18 2019

Status

approved