A286293 -id:A286293 - cp's OEIS frontend

A064736 a(1)=1, a(2)=2; for n>0, a(2n+2) = smallest number missing from {a(1), ... ,a(2n)}, and a(2n+1) = a(2n)a(2n+2).

1, 2, 6, 3, 12, 4, 20, 5, 35, 7, 56, 8, 72, 9, 90, 10, 110, 11, 143, 13, 182, 14, 210, 15, 240, 16, 272, 17, 306, 18, 342, 19, 399, 21, 462, 22, 506, 23, 552, 24, 600, 25, 650, 26, 702, 27, 756, 28, 812, 29, 870, 30, 930, 31, 992, 32, 1056, 33, 1122, 34, 1224, 36

Offset: 1

J. C. Lagarias (lagarias(AT)umich.edu), Oct 21 2001

Let c be the smallest positive constant such that for all permutations {a_n} of the positive integers, lim inf_{n -> infinity} gcd(a_n, a_{n+1})/n <= c. This sequence shows c >= 1/2.
The definition implies that if a(n) is prime then n is even. - N. J. A. Sloane, May 23 2017
a(2n) ~ n+1 ~ n has asymptotic density 1 and a(2n-1) ~ n(n+1) ~ n^2 has asymptotic density zero. - M. F. Hasler, May 23 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ray Chandler, Table of n, a(n) for n = 1..200000 (large file, 2.8 MB)
Ray Chandler, Table of n, a(n) for n = 1..2000000 (large gzipped file)
P. Erdős, R. Freud, and N. Hegyvári, Arithmetical properties of permutations of integers, Acta Mathematica Hungarica 41:1-2 (1983), pp. 169-176.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
Pierre Mazet, Eric Saias, Etude du graphe divisoriel 4, arXiv:1803.10073 [math.NT], 2018.
Index entries for sequences that are permutations of the natural numbers

A064745 gives inverse permutation.
Interleaving of A286290 and A286291. See also A286292, A286293.
Cf. A064764, A210770.

import Data.List (delete)
a064736 n = a064736_list !! (n-1)
a064736_list = 1 : 2 : f 1 2 [3..] where
   f u v (w:ws) = u' : w : f u' w (delete u' ws) where u' = v * w
-- Reinhard Zumkeller, Mar 23 2012

A064736 = {a[1]=1, a[2]=2}; a[n_] := a[n] = (an = If[OddQ[n], a[n-1]*a[n+1], First[ Complement[ Range[n], A064736]]]; AppendTo[A064736, an]; an); Table[a[n], {n, 1, 62}] (*Jean-François Alcover, Aug 07 2012 *)

More terms from Vladeta Jovovic, Oct 21 2001

Definition clarified by N. J. A. Sloane, May 23 2017

A286290 A bisection of A064736.

Original entry on oeis.org

1, 6, 12, 20, 35, 56, 72, 90, 110, 143, 182, 210, 240, 272, 306, 342, 399, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1224, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3135, 3306, 3422, 3540

Offset: 1

N. J. A. Sloane, May 23 2017

The terms of A064736 lie on two (curved) lines; this is one of them.
To produce this set, start with S={1} and a counter c=2, then repeatedly add to S the element c*increment(c), where increment() adds 1 or 2 in case c+1 is already in S. - M. F. Hasler, May 23 2017
Alternate definition: {1} and numbers of the form m(m+1) if neither m nor m+1 is an earlier term, or (m-1)(m+1), if m > 1 is a term of the sequence. - M. F. Hasler, May 23 2017
By definition, complement of A286291. - David A. Corneth, May 25 2017
If the initial 1 is omitted, this is the complement of A121229. - N. J. A. Sloane, May 26 2017

Ray Chandler, Table of n, a(n) for n = 1..10000
Ray Chandler, Table of n, a(n) for n = 1..1000000 (large gzipped file)

Cf. A064736, A286291, A286292, A286293, A121229.

A286290_list(Nmax,a=List(1),c=2)={while(#aM. F. Hasler, May 23 2017

a(n) = my(r = 1); for(i = 2, n, r = nxt(r)); r
is(n) = if(n < 6, return(n==1)); if(issquare(n+1, &n), is(n), if(sqrtint(4*n+1)^2 == 4*n+1, s = sqrtint(4*n+1); !(is(s\2) || is(s\2+1)), return(0)))
nxt(n) = n==1&&return(6); if(issquare(n+1, &n), (n+1) * (n+2), my(m = sqrtint(n)); if(is(m + 2), (m + 1) * (m + 3), (m + 1) * (m + 2)))
lista(n) = my(c = 1, l = List([1])); for(i=2, n, c = nxt(c); listput(l, c)); l \\ David A. Corneth, May 25 2017

a(n) ~ n^2*(1 + 1.5/n^c) with c=1/2. (Conjectured, although for small n around 10^5 a smaller c ~ 0.478 is a better fit to the data.) - M. F. Hasler, May 23 2017

For n around 10^8, c ~ 0.4848 is a better fit. - David A. Corneth, May 25 2017

A286291 A bisection of A064736.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, May 23 2017

nonn

The terms of A064736 lie on two (curved) lines; this is one of them.
Sequence is: a(1) = 2, a(2) = 3. m is in the sequence if and only if there is no i such that a(i) * a(i+1) = m, where i are indices of terms in the sequence so far. By definition, this is the complement of A286090. - David A. Corneth, May 25 2017
Apparently the same as A121229 shifted by one place. - R. J. Mathar, May 25 2017

			See comments: 4 is in the sequence, since the terms so far, 2 and 3, don't multiply to 4. Same for 5. Sequence so far is: 2, 3, 4, 5. 6 isn't in the sequence. 7 is. Carrying on we get 2, 3, 4, 5, 7, 8, 9, 10, 11. 12 isn't in the sequence. Further in the sequence, 30 is in the sequence though it's of the form k*(k+1) for k = 5. But 6 isn't in the sequence. And indeed, 5 and 7 are consecutive terms so 5*7 = 35 isn't in the sequence. - _David A. Corneth_, May 25 2017

Ray Chandler, Table of n, a(n) for n = 1..10000
Ray Chandler, Table of n, a(n) for n = 1..1000000 (large gzipped file)

Cf. A064736, A286290, A286292, A286293, A121229.

upto(n) = {my(l=List([2,3]), i = 1, p = 6, op = 3);
while(1, if(op>=n, return(l)); for(j=op + 1, p-1, listput(l, j)); i++; op = p; p = l[i]*l[i+1])}
is(n) = !is_A286290(n)
is_A286290(n) = if(n < 6, return(n==1)); if(issquare(n+1, &n), is(n), if(sqrtint(4*n+1)^2 == 4*n+1, s = sqrtint(4*n+1); !(is(s\2) || is(s\2+1)), return(0))) \\ David A. Corneth, May 25 2017

A286292 The first differences of A286291 (one of the bisections of A064736) appears to consist of runs of 1 followed by singleton 2's; this sequence gives the lengths of these runs.

Original entry on oeis.org

3, 4, 6, 13, 19, 14, 16, 18, 31, 37, 26, 28, 30, 32, 34, 55, 61, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 100, 106, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 163, 169, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138

Offset: 1

N. J. A. Sloane, May 23 2017

nonn

The steps of 2 occur when the corresponding integer is not in A286291 because it already occurred in A286290 [numbers of the form m(m+1) (m & m+1 not occurring earlier) or (m-1)(m+1) with m occurring earlier]. Accordingly, the present sequence equals first differences of A286290, minus 2. - M. F. Hasler, May 23 2017

			A064736: 1, 2, 6, 3, 12, 4, 20, 5, 35, 7, 56, 8, 72, 9, 90, 10, 110, ...
Bisect: 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, ... (A286291)
Differences: 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, ...
Runs: 3, 1, 4, 1, 6, 1, 13, 1, 19, 1, 14, 1, 16, 1, 18, 1, 31, 1, 37, 1, ...
Bisect: 3, 4, 6, 13, 19, 14, 16, 18, 31, 37, 26, 28, 30, 32, 34, 55, ... (this sequence)
From _M. F. Hasler_, May 23 2017: (Start)
Another approach:
A286290 = 1, 6, 12, 20, 35, 56, 72, 90, 110, 143, 182, 210, 240, 272, 306, 342, ...
1st Diff.: 5, 6,  8,  15, 21, 16, 18, 20,  33,  39,  28,  30,  32,  34, 36, ...
minus 2 =  3, 4,  6,  13, 19, 14, 16, 18,  31,  37,  26,  28,  30,  32, 34, ... (this sequence). (End)

Ray Chandler, Table of n, a(n) for n = 1..10000
Ray Chandler, Table of n, a(n) for n = 1..1000000 (large gzipped file)

Cf. A064736, A286290, A286291, A286293.

a(n) = A286290(n+1) - A286290(n) - 2. - M. F. Hasler, May 23 2017

cp's OEIS Frontend

A064736 a(1)=1, a(2)=2; for n>0, a(2*n+2) = smallest number missing from {a(1), ... ,a(2*n)}, and a(2*n+1) = a(2*n)*a(2*n+2).

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A286290 A bisection of A064736.

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A286291 A bisection of A064736.

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PARI

A286292 The first differences of A286291 (one of the bisections of A064736) appears to consist of runs of 1 followed by singleton 2's; this sequence gives the lengths of these runs.

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A064736 a(1)=1, a(2)=2; for n>0, a(2n+2) = smallest number missing from {a(1), ... ,a(2n)}, and a(2n+1) = a(2n)a(2n+2).