A286290 -id:A286290 - cp's OEIS frontend

A286293 A compressed version of A286290.

1, 2, 4, 11, 17, 12, 14, 16, 29, 35, 24, 26, 28, 30, 32, 53, 59, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 98, 104, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 161, 167, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136

Offset: 1

N. J. A. Sloane, May 23 2017

nonn

			Start with A064736, bisect to get A286290, take second difference, and we get:
1, 2, 7, 6, -5, 2, 2, 13, 6, -11, 2, 2, 2, 2, 21, 6, -19, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 36, 6, -34, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 57, 6, -55, 2, 2, ...
which appears to consist of runs of 2's separated by a triple of numbers.
Look at the runs in that sequence:
1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 11, 1, 1, 1, 17, 1, 1, 1, 12, 1, 1, 1, 14, 1, 1, 1, 16, 1, 1, 1, 29, 1, 1, 1, 35, 1, 1, 1, 24, 1, 1, 1, 26, 1, 1, ...
and take the 4i+2 subsequence, which gives the present sequence.

Ray Chandler, Table of n, a(n) for n = 1..971

Cf. A064736, A286290, A286291, A286292.

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).

Original entry on oeis.org

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

A121229 Beginning with a(1) = 1 and a(2) = 2, a(n) is not equal to the product of two consecutive (distinct) earlier terms.

Original entry on oeis.org

1, 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

Offset: 1

Giovanni Teofilatto, Aug 21 2006

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Cf. A005228, A030124.

The complement is A286290, excluding the initial 1.

A121229 := proc(n)
    option remember;
    local a,ispr,i;
    if n <=2 then
        n;
    else
        for a from procname(n-1)+1 do
            ispr := false ;
            for i from 1 to n-2 do
                if procname(i)*procname(i+1) = a then
                    ispr := true ;
                    break;
                end if;
            end do:
            if not ispr then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, May 25 2017

a[n_] := a[n] = Module[{k, ispr, i}, If[n <= 2, n, For[k = a[n - 1] + 1, True, k++, ispr = False; For[i = 1, i <= n - 2, i++, If[a[i]*a[i + 1] == k, ispr = True; Break[]]]; If[!ispr, Return[k]]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Sep 23 2022, after R. J. Mathar *)

from itertools import islice
def agen(): # generator of terms
    disallowed, prevk, k = {1, 2}, 2, 3; yield from [1, 2]
    while True:
        while k in disallowed: k += 1
        yield k; disallowed.update([k, k*prevk]); prevk = k
print(list(islice(agen(), 72))) # Michael S. Branicky, Sep 23 2022

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

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A286293 A compressed version of A286290.

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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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A121229 Beginning with a(1) = 1 and a(2) = 2, a(n) is not equal to the product of two consecutive (distinct) earlier terms.

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A286291 A bisection of A064736.

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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).