A249684 -id:A249684 - cp's OEIS frontend

A084937 Smallest number which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2.

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 15, 29, 14, 25, 27, 22, 31, 35, 12, 37, 41, 18, 43, 47, 20, 33, 49, 26, 45, 53, 28, 39, 55, 32, 51, 59, 38, 61, 63, 34, 65, 57, 44, 67, 69, 40, 71, 73, 24, 77, 79, 30, 83, 89, 36, 85, 91, 46, 75, 97, 52, 81

Offset: 1

Reinhard Zumkeller, Jun 13 2003

Equivalently, this is the lexicographically earliest sequence of positive numbers satisfying the condition that each term is relatively prime to the next two terms. - N. J. A. Sloane, Nov 03 2014
Empirically, the points lie roughly on two lines: if n == 2 mod 3 then a(n) ~= 2n/3, otherwise a(n) ~= 4n/3. See A249680-A249683 for the three trisections, and see also the Sigrist scatterplot. - N. J. A. Sloane, Nov 03 2014, Nov 04 2014
All primes and prime powers occur, and the primes occur in their natural order. For any prime p, p occurs before p^2 before p^3, ...
Empirically, this is a permutation of the natural numbers, with inverse A084933: a(A084933(n))=A084933(a(n))=n. It seems that there are no further fixed points after {1,2,3,8,33,39}. Empirically, a(n) mod 2 = A011655(n+1); ABS(a(n)-n) < n; a(3*n+1)>n; a(3*n+2)Reinhard Zumkeller, Dec 16 2007
For a(n) mod 3 see A249603. - N. J. A. Sloane, Nov 03 2014
A249694(n) = GCD(a(n),a(n+3)). - Reinhard Zumkeller, Nov 04 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..100000
John P. Linderman, Table of n, a(n) for n = 1..764179 (about 10MB)
Rémy Sigrist, Scatterplot of the first 2500 terms
Index entries for sequences that are permutations of the natural numbers

Cf. A084933 (inverse), A103683, A121216, A247665, A090252, A249603 (read mod 3), A249680, A249681, A249682, A249683 (trisections), A249694, A011655, A249684 (numbers that take a record number of steps to appear), A249685.
Indices of primes: A249602, and of prime powers: A249575.
Running counts of missing numbers: A249686, A250099, A250100; A249777, A249856, A249857.
Where a(3n)>a(3n+1): A249689.
See also A353706, A353709, A353710.

import Data.List (delete)
a084937 n = a084937_list !! (n-1)
a084937_list = 1 : 2 : f 2 1 [3..] where
   f x y zs = g zs where
      g (u:us) | gcd y u > 1 || gcd x u > 1 = g us
               | otherwise = u : f u x (delete u zs)
-- Reinhard Zumkeller, Jan 28 2012

N:= 1000: # to get a(n) until the first entry > N
a[1]:= 1: a[2]:= 2:
R:= {$3..N}:
for n from 3 while R <> {} do
  success:= false;
  for r in R do
    if igcd(r,a[n-1]) = 1 and igcd(r,a[n-2])=1 then
       a[n]:= r;
       R:= R minus {r};
       success:= true;
       break
    fi
  od:
  if not success then break fi;
od:
seq(a[i], i = 1 .. n-1); # Robert Israel, Dec 12 2014

lst={1,2,3}; unused=Range[4,100]; While[n=Select[unused, CoprimeQ[#, lst[[-1]]] && CoprimeQ[#, lst[[-2]]] &, 1]; n != {}, AppendTo[lst, n[[1]]]; unused=DeleteCases[unused, n[[1]]]]; lst
f[s_] := Block[{k = 1, l = Take[s, -2]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Nest[f, {1, 2}, 67] (* Robert G. Wilson v, Jun 26 2011 *)

taken(k,t=v[k])=for(i=3,k-1, if(v[i]==t, return(1))); 0
step(k,g)=while(gcd(k,g)>1, k++); k
first(n)=local(v=vector(n,i,i)); my(nxt=3,t); for(k=3,n, v[k]=step(nxt, t=v[k-1]*v[k-2]); while(taken(k), v[k]=step(v[k]+1,t)); if(v[k]==t, while(taken(k+1,t++),))); v \\ Charles R Greathouse IV, Aug 26 2016

from math import gcd
A084937_list, l1, l2, s, b = [1,2], 2, 1, 3, set()
for _ in range(10**3):
    i = s
    while True:
        if not i in b and gcd(i,l1) == 1 and gcd(i,l2) == 1:
            A084937_list.append(i)
            l2, l1 = l1, i
            b.add(i)
            while s in b:
                b.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 09 2014

Entry revised by N. J. A. Sloane, Nov 04 2014

A249777 Let z = A084937: a(n) = number of numbers <= z(n) that are != z(k) for k=1..n-1 and not coprime to z(n-1) and z(n-2).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 2, 3, 0, 5, 6, 0, 6, 7, 3, 2, 10, 1, 5, 6, 3, 7, 10, 0, 10, 13, 0, 13, 16, 0, 5, 15, 1, 11, 16, 1, 6, 15, 2, 11, 16, 4, 16, 17, 2, 17, 12, 5, 16, 17, 3, 17, 18, 0, 20, 21, 0, 23, 28, 0, 23, 27, 1, 15, 30, 3, 17, 26, 4, 19, 28, 2, 21

Offset: 1

Reinhard Zumkeller, Nov 05 2014

nonn

a(A249684(n)) = 0.

			.   n | A084937(n) |    unused numbers less than A084937(n)   | a(n)
. ----+------------+------------------------------------------+-----
.   3 |          3 |  _                                       |    0
.   4 |          5 |  4                                       |    1
.   5 |          4 |  _                                       |    0
.   6 |          7 |  6                                       |    1
.   7 |          9 |  6, 8                                    |    2
.   8 |          8 |  6                                       |    1
.   9 |         11 |  6, 10                                   |    2
.  10 |         13 |  6, 10, 12                               |    3
.  11 |          6 |  _                                       |    0
.  12 |         17 |  10, 12, 14, 15, 16                      |    5
.  13 |         19 |  10, 12, 14, 15, 16, 18                  |    6
.  14 |         10 |  _                                       |    0
.  15 |         21 |  12, 14, 15, 16, 18, 20                  |    6
.  16 |         23 |  12, 14, 15, 16, 18, 20, 22              |    7
.  17 |         16 |  12, 14, 15                              |    3
.  18 |         15 |  12, 14                                  |    2
.  19 |         29 |  12, 14, 18, 20, 22, 24, 25, 26, 27, 28  |   10
.  20 |         14 |  12                                      |    1
.  21 |         25 |  12, 18, 20, 22, 24                      |    5
.  22 |         27 |  12, 18, 20, 22, 24, 26                  |    6
.  23 |         22 |  12, 18, 20                              |    3
.  24 |         31 |  12, 18, 20, 24, 26, 28, 30              |    7
.  25 |         35 |  12, 18, 20, 24, 26, 28, 30, 32, 33, 34  |   10 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A084937, A249684.

a249777 n = a249777_list !! (n-1)
a249777_list = 0 : 0 : f 2 1 [3..] where
   f x y zs = g zs 0 where
      g (u:us) w | gcd y u > 1 || gcd x u > 1 = g us (w + 1)
                 | otherwise = w : f u x (delete u zs)

A084933 Inverse of A084937.

Original entry on oeis.org

1, 2, 3, 5, 4, 11, 6, 8, 7, 14, 9, 26, 10, 20, 18, 17, 12, 29, 13, 32, 15, 23, 16, 56, 21, 35, 22, 38, 19, 59, 24, 41, 33, 47, 25, 62, 27, 44, 39, 53, 28, 95, 30, 50, 36, 65, 31, 92, 34, 74, 42, 68, 37, 98, 40, 71, 49, 77, 43, 101, 45, 80, 46, 83, 48, 137, 51, 86, 52

Offset: 1

Reinhard Zumkeller, Jun 13 2003

nonn

Empirically, A084937 is a permutation of the natural numbers. Assuming that is true, we will have a(A084937(n)) = A084937(a(n)) = n. (Comment revised by N. J. A. Sloane, Nov 05 2014)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A084937. For records see A249684, A249685.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a084933 n = (fromJust $ elemIndex n a084937_list) + 1
-- Reinhard Zumkeller, Jan 28 2012

f[s_] := Block[{k = 1, l = Take[s, -2]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Ordering@ Nest[f, {1, 2}, 100] (* Robert G. Wilson v, Jun 26 2011 *)

cp's OEIS Frontend

A249685 Indices where terms of A249684 appear in A084933.

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A084937 Smallest number which is coprime to the last two predecessors and has not yet appeared; a(1)=1, a(2)=2.

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A249777 Let z = A084937: a(n) = number of numbers <= z(n) that are != z(k) for k=1..n-1 and not coprime to z(n-1) and z(n-2).

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A084933 Inverse of A084937.

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