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A004790 a(n) = n-th number k >= 2 such that if 1 < j < k then fractional part of log k < fractional part of log j. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 20 19:36 EDT 2019. Contains 324234 sequences. (Running on oeis4.)