A004790 Numbers k >= 2 such that if 1 < j < k then (fractional part of log k) < (fractional part of log j).

2, 3, 8, 21, 55, 149, 404, 1097, 2981, 162755, 1202605, 3269018, 8886111, 24154953, 178482301, 9744803447, 26489122130, 195729609429, 532048240602, 1446257064292, 3931334297145, 10686474581525, 29048849665248, 78962960182681, 583461742527455, 1586013452313431

Offset

1,1

Comments

Sequence lists all numbers k > 1 for which the fractional part of log(k) reaches a record low. For n > 1, this can happen only when a(n) = ceiling(e^m) for some positive integer m; see Example section. - Jon E. Schoenfield, May 28 2018

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Example

From Jon E. Schoenfield, May 28 2018: (Start)
k = ceiling(e^m) yields a term for some but not all positive integers m:
.
   m |      k=ceiling(e^m)       |          log(k)
  ---+---------------------------+--------------------------
   1 |                 3 = a(2)  |  1.0986122886681096913...
   2 |                 8 = a(3)  |  2.0794415416798359282...
   3 |                21 = a(4)  |  3.0445224377234229965...
   4 |                55 = a(5)  |  4.0073331852324709186...
   5 |               149 = a(6)  |  5.0039463059454591409...
   6 |               404 = a(7)  |  6.0014148779611500697...
   7 |              1097 = a(8)  |  7.0003344602752302459...
   8 |              2981 = a(9)  |  8.0000140936780714441...
   9 |              8104         |  9.0001130459285193087...
  10 |             22027         | 10.0000242525841575280...
  11 |             59875         | 11.0000143347132163589...
  12 |            162755 = a(10) | 12.0000012815651115743...
  13 |            442414         | 13.0000013742591718739...
  14 |           1202605 = a(11) | 14.0000005952373691014...
  15 |           3269018 = a(12) | 15.0000001919622191103...
  16 |           8886111 = a(13) | 16.0000000539597288735...
  17 |          24154953 = a(14) | 17.0000000102018291255...
  18 |          65659970         | 18.0000000131384387554...
  19 |         178482301 = a(15) | 19.0000000002062542837...
  20 |         485165196         | 20.0000000012165129058...
  21 |        1318815735         | 21.0000000003918555785...
  22 |        3584912847         | 22.0000000002422397629...
  23 |        9744803447 = a(16) | 23.0000000000770767110...
  24 |       26489122130 = a(17) | 24.0000000000059091314...
  25 |       72004899338         | 25.0000000000085289679...
  26 |      195729609429 = a(18) | 26.0000000000008237677...
  27 |      532048240602 = a(19) | 27.0000000000003785057...
  28 |     1446257064292 = a(20) | 28.0000000000003628859...
  29 |     3931334297145 = a(21) | 29.0000000000002436642...
  30 |    10686474581525 = a(22) | 30.0000000000000503302...
  31 |    29048849665248 = a(23) | 31.0000000000000197862...
  32 |    78962960182681 = a(24) | 32.0000000000000038605...
  33 |   214643579785917         | 33.0000000000000043578...
  34 |   583461742527455 = a(25) | 34.0000000000000002032...
  35 |  1586013452313431 = a(26) | 35.0000000000000001714...
  36 |  4311231547115196         | 36.0000000000000001792...
.
For k = ceiling(e^m) > 2, 0 < frac(log(k)) < e^(-m), so frac(log(k)) must approach 0 as m increases, but it does not do so monotonically; at values of m where frac(log(k)) is particularly small relative to e^(-m) (e.g., at m = 8 or m = 19), the next term after a(n) = k = ceiling(e^m) can be as large as a(n+1) = ceiling(e^(ceiling(-log(frac(log(k)))))).
(End)

Prog

(PARI) lista(n) = {last = frac(log(2)); for (k=2, n, new = frac(log(k)); if (new < last, print1 (k, ", "); last = new; ); ); } \\ Michel Marcus, Mar 21 2013

Crossrefs

Cf. A004791.

Sequence in context: A288252 A122263 A132730 * A245464 A243562 A238329

Adjacent sequences: A004787 A004788 A004789 * A004791 A004792 A004793

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from David W. Wilson

a(24)-a(26) from Jon E. Schoenfield, May 28 2018

Status

approved