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A139080
a(n) = the smallest positive integer not occurring earlier in the sequence such that floor(max(a(n),a(n-1))/min(a(n),a(n-1))) = 2; a(1) = 1.
3
1, 2, 4, 8, 3, 6, 12, 5, 10, 20, 7, 14, 28, 11, 22, 9, 18, 36, 13, 26, 52, 19, 38, 15, 30, 60, 21, 42, 16, 32, 64, 23, 46, 17, 34, 68, 24, 48, 96, 33, 66, 25, 50, 100, 35, 70, 27, 54, 108, 37, 74, 29, 58, 116, 39, 78, 31, 62, 124, 43, 86, 40, 80, 160, 55, 110, 41, 82, 164, 56
OFFSET
1,2
COMMENTS
Is there always an unused positive integer, a(n), such that floor(max(a(n),a(n-1))/min(a(n),a(n-1))) = 2, or does the sequence terminate? If the sequence is infinite, is it a permutation of the positive integers?
The sequence grows in 5 distinct "bands" which are populated respectively by every 5th term; these terms are in proportion (a(5k), a(5k+1), a(5k+2), a(5k+3), a(5k+4)) ~ (4, 8, 3, 6, 12)*k*(1-O(1/log(k))). More precisely, for k>6, we have a(5k+1) = 2 a(5k), a(5k+2) ~ (3/8)*a(5k+1) is the least number not occurring earlier, a(5k+3) = 2 a(5k+2), a(5k+4) = 2 a(5k+3) and a(5k+5) is the least number greater than a(5k+4)/3 not occurring earlier. - M. F. Hasler, Apr 24 2015
See LINKS for a proof of the preceding statement, which provides an affirmative answer to the questions raised in the first comment. See A257280 for the inverse permutation. See also the variant A257470 and its inverse A257271. - M. F. Hasler, Apr 26 2015
LINKS
M. F. Hasler, A139080, OEIS wiki, April 2015.
FORMULA
For k>6, when n = 5k, 5k+2 or 5k+3, then a(n+1) = 2 a(n); and with N(n) := N \ {a(k); k < n}, a(5k+2) = min N(5k+2) and a(5k+5) = min { m in N(5k+5), m > a(5k+4)/3 }. - M. F. Hasler, Apr 26 2015
MATHEMATICA
f[n_] := Block[{s = {1}, i, k}, For[i = 2, i <= n, i++, k = 1; While[Nand[! MemberQ[s, k], Floor[Max[k, s[[i - 1]]]/Min[k, s[[i - 1]]]] == 2], k++]; AppendTo[s, k]]; s]; f@ 70 (* Michael De Vlieger, Apr 26 2015 *)
PROG
(PARI) { t=0; last=1; for( n=1, 10000, write("b139080.txt", n, " ", last); t+=1<<last; for( i=last\3+1, last\2, bittest(t, i) & next; last=i; next(2)); for( i=last*2, last*3-1, bittest(t, i) & next; last=i; next(2)); error("THE END: n=", n)); print("Largest term used:"); log(t)\log(2)} \\ M. F. Hasler, Apr 07 2008
(Haskell)
import Data.List (delete)
a139080 n = a139080_list !! (n-1)
a139080_list = 1 : f 1 [2..] where
f x zs = g zs where
g (y:ys) = if x < y && y `div` x == 2 || x `div` y == 2
then y : f y (delete y zs) else g ys
-- Reinhard Zumkeller, Mar 14 2014
CROSSREFS
Cf. A257280 (inverse), A257470, A257271.
Sequence in context: A232645 A378376 A257470 * A036118 A247555 A340730
KEYWORD
nonn,look
AUTHOR
Leroy Quet, Apr 07 2008
EXTENSIONS
Additional terms calculated by Robert Israel and M. F. Hasler, Apr 11 2008
STATUS
approved