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
Keywords
Examples
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)
Links
- Jon E. Schoenfield, Table of n, a(n) for n = 1..100
Crossrefs
Cf. A004791.
Programs
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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
Extensions
More terms from David W. Wilson
a(24)-a(26) from Jon E. Schoenfield, May 28 2018
Comments