A297840 Numbers k > 0 that set a new record for the closeness of 4*Pi*k^2 to an integer.

1, 2, 3, 4, 14, 99, 507, 5112, 9361, 13451, 90425, 132640, 268883, 462518, 1803181, 1890795, 2053555, 3831113, 4166332, 5759263, 38574916, 45164470, 310321816, 530684437

Offset

1,2

Comments

Integer radii such that the surface area of the corresponding sphere is closer to an integer than for any smaller integer radius.

Links

Table of n, a(n) for n=1..24.

Example

          k |                 4*Pi*k^2              | Deviation from
            |                                       | integer
------------+---------------------------------------+----------------------
          1 |                  12.56637061435917... | 0.43362938564082...
          2 |                  50.26548245743669... | 0.26548245743669...
          3 |                 113.09733552923255... | 0.09733552923255...
          4 |                 201.06192982974676... | 0.06192982974676...
         14 |                2463.00864041439789... | 0.00864041439789...
         99 |              123162.99839133425412... | 0.00160866574587...
        507 |             3230173.00005041104861... | 0.00005041104861...
       5112 |           328391233.00004811902011... | 0.00004811902011...
       9361 |          1101169958.00003281689453... | 0.00003281689453...
      13451 |          2273625908.00000716139558... | 0.00000716139558...
      90425 |        102751199128.99999628277400... | 0.00000371722599...
     132640 |        221084802748.99999692741688... | 0.00000307258311...
     268883 |        908524313282.00000157554683... | 0.00000157554683...
     462518 |       2688234448369.99999894165289... | 0.00000105834710...
    1803181 |      40859072996351.99999911345115... | 0.00000088654884...
    1890795 |      44926103614145.99999944953623... | 0.00000055046376...
    2053555 |      52993492455840.00000053265439... | 0.00000053265439...
    3831113 |     184441985069785.99999958888834... | 0.00000041111165...
    4166332 |     218131111695367.00000020961660... | 0.00000020961660...
    5759263 |     416815333018180.99999995070232... | 0.00000004929767...
   38574916 |   18699062881733779.00000003869142... | 0.00000003869142...
   45164470 |   25633251606933903.00000000438530... | 0.00000000438530...
  310321816 | 1210136834140739074.00000000262227... | 0.00000000262227...
  530684437 | 3539016334684589995.00000000014286... | 0.00000000014286...

Mathematica

mx = 1; k = 1; lst = {}; While[k < 3000000001, a = N[ Pi(2k)^2, 32]; a = N[ Abs[a - Round@ a], 32]; If[a < mx, mx = a; AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Jan 11 2018 *)

Prog

(PARI) closeness(n) = my(s=4*Pi*n^2); if(round(s) > s, return(round(s)-s), return(s-round(s)))
my(r=1, k=1, c=0); while(1, c=closeness(k); if(c < r, print1(k, ", "); r=c); k++)

Crossrefs

Cf. A066644, A135971, A254714, A297839.

Sequence in context: A280924 A295756 A140047 * A102483 A134916 A085100

Adjacent sequences: A297837 A297838 A297839 * A297841 A297842 A297843

Keyword

nonn,hard,more

Author

Felix Fröhlich, Jan 07 2018

Extensions

a(23)-a(24) from Jon E. Schoenfield, Jan 07 2018

Status

approved