A121216 -id:A121216 - cp's OEIS frontend

A377091 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a square; in case of a tie, preference is given to the positive value.

0, 1, 2, -2, -1, 3, 4, 5, -4, -3, 6, 7, 8, -8, -7, -6, -5, -9, -10, -11, -12, 13, 9, 10, 11, 12, -13, -14, -15, -16, -17, -18, 18, 14, 15, 16, 17, -19, -20, -21, -22, -23, -24, 25, 21, 20, 19, 23, 22, 26, 27, 28, 24, -25, -26, -27, -28, -29, -30, -31, -32, 32

Offset: 0

Rémy Sigrist, Oct 16 2024

Conjecture 1: Every integer (positive or negative) appears in this sequence.
Conjecture 2: For n > 16, |a(n)| is within sqrt(n/2) of floor(n/2). See A379071. - N. J. A. Sloane, Dec 29 2024 [Corrected by Paolo Xausa, Jan 21 2025]
Conjecture 3: lim sup ||a(n)| - floor(n/2)|/sqrt(n) = 1/2. (See link.) - N. J. A. Sloane and Paolo Xausa, Feb 03 2025
Conjecture 4: After a(n) has been found, the sequence contains all numbers in the range [0,f(n)], where lim sup f(n) = (n-sqrt(n))/2. There is a corresponding conjecture for the negative terms. See A379067. - N. J. A. Sloane and Paolo Xausa, Feb 13 2025

			The initial terms are:
  n   a(n)  |a(n)-a(n-1)|
  --  ----  -------------
   0     0  N/A
   1     1  1^2
   2     2  1^2
   3    -2  2^2
   4    -1  1^2
   5     3  2^2
   6     4  1^2
   7     5  1^2
   8    -4  3^2
   9    -3  1^2
  10     6  3^2
  11     7  1^2
  12     8  1^2
  13    -8  4^2
  14    -7  1^2

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [First 10000 terms from Rémy Sigrist]
M. F. Hasler, Interactive spiral graph representation of the first n terms, Jan 18 2025.
Kerry Mitchell, Spiral graph representation of first 31 terms
Kerry Mitchell, Colored spiral graph representation of first 100 terms
Chris Scussel, Spiral graph representation of 5000 terms
Rémy Sigrist, Scatterplot of the first 10000 terms of the partial sums
Rémy Sigrist, Christmas card based on graph of the partial sums [Rotate graph by 90 degs, add color and decorations]
Rémy Sigrist, PARI program
N. J. A. Sloane, Table of n, a(n) for n = 0..250000
N. J. A. Sloane, Spiral graph representation of first 28 terms [Hand-drawn]
Paolo Xausa, Scatterplot of ||a(n)| - floor(n/2)| for 0 <= n <= 200000. The orange line is sqrt(n)/2.

This sequence is a variant of A277616 allowing negative values.
Cf. A277617, A277618, A377090, A377092.
A large number of sequences have been derived from the present sequence in the hope (so far unfulfilled) of finding a formula or recurrence: see A379057-A379078, A379786-A379798, A379802, A379803, A379804, A379880, A380223, A380224, A380225, A382715-A382718.
First differences are A379061 (certainly the most relevant derived sequence). - M. F. Hasler, Feb 08 2025
"Lexicographically earliest" sequences for which there is a proof that every number that could appear does appear: A064413, A098550, A109812, A121216, A347113, etc. - N. J. A. Sloane, Feb 08 2025

A377091 = [0]; A377091.least_unused = 1;
function a(n){
  for(let i = A377091.length-1; i < n; ++i) {
    let k = A377091.least_unused;
    while(!Number.isInteger(Math.sqrt(Math.abs(A377091[i] - k)))
          || A377091.indexOf(k) > 0) k = (k<0)-k;
    A377091.push(k);
    if (k == A377091.least_unused) {
      do k = (k<0)-k; while ( A377091.indexOf( k ) > 0 );
      A377091.least_unused = k;
  } };
  return A377091[n];
} // M. F. Hasler, Jan 26 2025

h := proc(b, a, i) option remember; ifelse(issqr(abs(a[-1] - i)) and not is(i in a), ifelse(b < nops(a) + 1, a, h(b, [op(a), i], 1)), h(b, a, ifelse(i < 0, 1 - i, -i))) end:
a_list := length -> h(length, [0], 1): a_list(62);  # Peter Luschny, Jan 20 2025

A377091list[nmax_] := Module[{s, a, u = 1}, s[_] := False; s[0] = True; NestList[(While[s[u] && s[-u], u++]; a = u; While[s[a] || !IntegerQ[Sqrt[Abs[# - a]]], a = Boole[a < 0] - a]; s[a] = True; a) &, 0,nmax]];
A377091list[100] (* Paolo Xausa, Mar 18 2025 *)

PARI
```
\\ See Links section.
```

A377091_upto(n,S=[])={vector(n+1, k, S=setunion(S, [n=if(k>1, k=1; while(setsearch(S,k) || !issquare(abs(n-k)), k=(k<0)-k); k)]); n)} \\ M. F. Hasler, Jan 18 2025

from math import isqrt
from itertools import count, islice
def cond(n): return isqrt(n)**2 == n
def agen(): # generator of terms
    an, aset, m = 0, {0}, 1
    for n in count(0):
        yield an
        an = next(s for k in count(m) for s in [k, -k] if s not in aset and cond(abs(an-s)))
        aset.add(an)
        while m in aset and -m in aset: m += 1
print(list(islice(agen(), 62))) # Michael S. Branicky, Dec 25 2024

from math import sqrt
def a_list(b: int, a: list[int] = [0], i: int = 1) -> list[int]:
    if sqrt(abs(a[-1] - i)).is_integer() and not (i in a):
        a += [i]
        if b < len(a):
            return a
        else:
            return a_list(b, a)
    else:
        return a_list(b, a, int(i < 0) - i)
print(a_list(40))  # Peter Luschny, Jan 20 2025

class A377091: # A377091(n) gives a(n)
    terms = [0]; N = 1 # next candidate
    def _new_(A, n): A.extend(A, n-len(A.terms)+1); return A.terms[n]
    def extend(A, n): any((k:=A.N) in A.terms and setattr(A, 'N', k:=(k<0)-k) or
        A.terms.append(next(k for _ in range(9**9) if (abs(A.terms[-1]-k)**.5)
       .is_integer() and k not in A.terms or not(k:=(k<0)-k))) for _ in range(n))
# M. F. Hasler, Feb 08 2025

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

Original entry on oeis.org

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

A353709 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 12, 32, 17, 6, 40, 64, 5, 10, 48, 65, 14, 128, 33, 18, 68, 9, 34, 20, 72, 35, 132, 24, 66, 36, 25, 130, 96, 13, 144, 98, 256, 21, 42, 192, 257, 22, 104, 129, 258, 28, 97, 384, 26, 37, 320, 136, 7, 80, 160, 11, 84, 288, 131, 76, 272, 161, 70, 264, 49, 134, 328, 512, 19, 44, 448, 513, 30, 224, 768, 15, 112, 640, 259, 52, 200, 514, 53

Offset: 0

N. J. A. Sloane, May 06 2022

A set-theory analog of A084937.

Conjecture: This is a permutation of the nonnegative numbers.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Walter Trump, Table of n, a(n) for n = 0..10^6
Walter Trump, Log-log plot of first 2^24 terms

Cf. A084937 (number theory analog), A109812, A121216, A353405 (powers of 2), A353708, A353710, A353715 and A353716 (a(n)+a(n+1)), A353717 (inverse), A353718, A353719 (primes), A353720 and A353721 (Records).

For the numbers that are the slowest to appear see A353723 and A353722.

read(transforms) : # ANDnos def'd here
A353709 := proc(n)
option remember;
local c, i, known ;
if n <= 2 then
n;
else
for c from 1 do
known := false ;
for i from 1 to n-1 do
if procname(i) = c then
known := true;
break ;
end if;
end do:
if not known and ANDnos(c, procname(n-2)) = 0 and ANDnos(c, procname(n-1)) = 0 then
return c;
end if;
end do:
end if;
end proc: # Following R. J. Mathar's program for A109812.
[seq(A353709(n), n=0..256)] ;
# second Maple program:
b:= proc() false end: t:= 2:
a:= proc(n) option remember; global t; local k; if n<2 then n
      else for k from t while b(k) or Bits[And](k, a(n-2))>0
      or Bits[And](k, a(n-1))>0 do od; b(k):=true;
      while b(t) do t:=t+1 od; k fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 06 2022

nn = 83; c[] = -1; a[0] = c[0] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[a[n - 1], k] == 0, BitAnd[a[n - 2], k] == 0], k++]; Set[{a[n], c[k]}, {k, n}]; If[k == u, While[c[u] > -1, u++]], {n, 2, nn}]; Array[a, nn+1, 0] (* _Michael De Vlieger, May 06 2022 *)

from itertools import count, islice
def A353709_gen(): # generator of terms
    s, a, b, c, ab = {0,1}, 0, 1, 2, 1
    yield from (0,1)
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                yield n
                a, b = b, n
                ab = a|b
                s.add(n)
                while c in s:
                    c += 1
                break
A353709_list = list(islice(A353709_gen(),20)) # Chai Wah Wu, May 07 2022

A354169 a(0) = 0, a(1) = 1, a(2) = 2; for k >= 2, given a(k), the sequence is extended by adjoining two terms: a(2k-1) = smallest m >= 0 not among a(0) .. a(k) such that {m, a(k), a(k+1), ..., a(2k-2)} are pairwise disjoint in binary, and a(2k) = smallest m >= 0 not among a(0) .. a(k) such that {m, a(k), ..., a(2k-1)} are pairwise disjoint in binary.

Original entry on oeis.org

0, 1, 2, 4, 8, 3, 16, 32, 64, 12, 128, 256, 512, 17, 1024, 34, 2048, 4096, 8192, 68, 16384, 136, 32768, 65536, 131072, 768, 262144, 524288, 1048576, 1025, 2097152, 18, 4194304, 2080, 8388608, 16777216, 33554432, 12288, 67108864, 134217728, 268435456, 16388

Offset: 0

N. J. A. Sloane, Jun 05 2022

The paper by De Vlieger et al. (2022) calls this the "binary two-up sequence".
"Pairwise disjoint in binary" means no common 1-bits in their binary representations.
This is a set-theory analog of A090252. It bears the same relation to A090252 as A252867 does to A098550, A353708 to A121216, A353712 to A347113, etc.
A consequence of the definition, and also an equivalent definition, is that this is the lexicographically earliest infinite sequence of distinct nonnegative numbers with the property that the binary representation of a(n) is disjoint from (has no common 1's with) the binary representations of the following n terms.
An equivalent definition is that a(n) is the smallest nonnegative number that is disjoint (in its binary representation) from each of the previous floor(n/2) terms.
For the subsequence 0, 3, 12, 17, 34, ... of the terms that are not powers of 2 see A354680 and A354798.
All terms are the sum of at most two powers of 2 (see De Vlieger et al., 2022). - N. J. A. Sloane, Aug 29 2022

			After a(2) = 2 = 10_2, a(3) must equal ?0?_2, and the smallest such number we have not seen is a(3) = 100_2 = 4, and a(4) must equal ?00?_2, and the smallest such number we have not seen is a(4) = 1000_2 = 8.

Rémy Sigrist, Table of n, a(n) for n = 0..4941
Michael De Vlieger, Thomas Scheuerle, Rémy Sigrist, N. J. A. Sloane, and Walter Trump, The Binary Two-Up Sequence, arXiv:2209.04108 [math.CO], Sep 11 2022.
Rémy Sigrist, PARI program
N. J. A. Sloane, Construction of terms a(0) through a(18)
N. J. A. Sloane, Table showing first 190 terms (pdf file of scan of large table)
N. J. A. Sloane, Left-hand portion of rows 97-135 of preceding table, showing columns 67-96, rotated counter-clockwise by 90 degrees. The red line in this table should be aligned with the red line between rows 135 and 136 in the preceding table. [This portion of the table was too wide to fit through the scanner.]
N. J. A. Sloane, Blog post about the Two-Up sequence, June 13 2022. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 16.

A355889 is a more efficient way to present this sequence.
Cf. A000120, A090252, A098550, A121216, A252867, A347113, A353708, A353712, A354680, A354767, A354798, A355150.
See also A354773, A354774, A354775, A354780, A354781, A354783.

nn = 42; c[] = r = 0; m = 1; Array[Set[{a[#], c[#]}, {#, #}] &, 2, 0]; Do[k = SelectFirst[Union@ Map[Total, Rest@ Subsets[2^Reverse[Length[#] - Position[#, 1][[All, 1]]] &@ IntegerDigits[2^(r + 2) - m - 1, 2]]], c[#] == 0 &]; Set[{a[n], c[k]}, {k, n}]; m += a[n]; If[And[IntegerQ[#], # > 0], m -= a[#]] &[n/2]; If[And[EvenQ[k], PrimePowerQ[k], k > 2^r], r++], {n, 2, nn}]; Table[a[n], {n, 0, nn}] (* _Michael De Vlieger, Jul 13 2022 *)

PARI
```
See Links section.
```

from itertools import count, islice
from collections import deque
from functools import reduce
from operator import or_
def A354169_gen(): # generator of terms
    aset, aqueue, b, f = {0,1,2}, deque([2]), 2, False
    yield from (0,1,2)
    while True:
        for k in count(1):
            m, j, j2, r, s = 0, 0, 1, b, k
            while r > 0:
                r, q = divmod(r,2)
                if not q:
                    s, y = divmod(s,2)
                    m += y*j2
                j += 1
                j2 *= 2
            if s > 0:
                m += s*2**b.bit_length()
            if m not in aset:
                yield m
                aset.add(m)
                aqueue.append(m)
                if f: aqueue.popleft()
                b = reduce(or_,aqueue)
                f = not f
                break
A354169_list = list(islice(A354169_gen(),40)) # Chai Wah Wu, Jun 06 2022

More terms from Rémy Sigrist, Jun 06 2022

A270139 a(n)=n when n<=3, otherwise a(n) is the smallest unused positive integer which is not coprime to the two previous terms.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 15, 10, 5, 20, 25, 30, 35, 14, 7, 21, 28, 18, 4, 8, 16, 22, 24, 26, 32, 34, 36, 38, 40, 42, 44, 33, 11, 55, 66, 45, 27, 39, 48, 51, 54, 57, 60, 63, 56, 49, 70, 77, 84, 88, 46, 50, 52, 58, 62, 64, 68, 72, 74, 76, 78, 80, 65, 75, 85, 90, 95, 100, 105, 96

Offset: 1

Ivan Neretin, Mar 11 2016

nonn

Other possible conditions on a(n) with respect to its common factors with a(n-2) and a(n-1) lead to the following:
Coprime to both: A084937.
Coprime to the latter and not the former: A098550.
Coprime to the former and not the latter: with any initial conditions, the sequence "paints itself into a corner", i.e., is finite. With the added condition of a(n) having an extra prime factor not contained in a(n-1), it is A336957.
Coprime to the latter, regardless of the former: simply A000027.
Coprime to the former, regardless of the latter: A121216.
Non-coprime to the latter, regardless of the former: A064413.
Non-coprime to the former, regardless of the latter: A121217.

			a(12) = 30, a(13) = 35, so a(14) must have common factors (possibly different) with 30 and 35, and the smallest unused number with that property turns out to be 14, so a(14) = 14.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A064413, A084937, A098550, A121216, A121217, A254077, A255582, A336957.

a = {1, 2, 3}; Do[k = 1; While[(MemberQ[a, k] || GCD[a[[-1]], k] == 1 || GCD[a[[-2]], k] == 1), k++]; AppendTo[a, k], {n, 2, 68}]; a

A352588 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) but does not equal a(n-1)+1.

Original entry on oeis.org

1, 2, 5, 3, 7, 4, 9, 8, 11, 6, 13, 10, 17, 12, 19, 14, 23, 15, 22, 21, 16, 25, 18, 29, 20, 27, 26, 31, 24, 35, 32, 37, 28, 33, 38, 41, 30, 43, 34, 39, 44, 47, 36, 49, 40, 51, 46, 45, 52, 55, 42, 53, 48, 59, 50, 57, 56, 61, 54, 65, 58, 63, 62, 67, 60, 71, 64, 69, 68, 73, 66, 79, 70, 81, 74, 77

Offset: 1

Scott R. Shannon, Apr 29 2022

nonn

Theorem: This is a permutation of the natural numbers. The proof is essentially the same as for A093714. - N. J. A. Sloane, May 02 2022
Coincides with A093714 for n >= 17. - Scott R. Shannon, May 02 2022.
In the first 100 million terms the sequence's values are concentrated along the line a(n) = n, resulting in 1160 fixed points in this range. However the last fixed point in this range is a(1034312), with the sequence oscillating above and below the line a(n) = n from then on. It is unknown if this behavior continues or if more fixed points eventually appear.
The largest offset in the first 100 million terms from the line a(n) = n is a(45902952) = 45902981, with an offset of 29. In this range a number is rejected as the next term on 207 occasions as it equals a(n-1)+1.
Beyond a(4) = 3 the primes appear in their natural order.
From Michael De Vlieger, May 01 2022: (Start)
Theorem: if prime p | a(n-1) then p does not divide a(n). Proof: primes either divide or are coprime to a given number. We say numbers m and n are coprime iff gcd(m,n) = 1. Suppose p | a(n-1) and p | a(n), then p | m, where m = gcd(a(n-1), a(n)). By definition of prime and divisor, m > 1, a contradiction.
Corollary: even terms do not appear adjacently in the sequence, however we may have runs of odd terms.
Theorem: fixed point a(n) = n implies a(n) and n have the same parity. Proof: a(n) = n iff a(n) mod n = 0, since n | n. Suppose prime q|n yet gcd(a(n), q) = 1, then a(n) != n, a contradiction.
Observation: there are 9 runs of odd terms for n = 1..2^28, one of 3 odd terms {5, 3, 7}, the rest of 2. Fixed points appear in intervals [1, 3], [4, 17], [78, 1787], [15022, 38123], and [45053, 1036043]. The last run of odd terms for n <= 2^28 begins at n = 1036043. Is there another run of odd terms that will begin a new interval that harbors fixed points? (End)

			a(3) = 5 as a(2) = 2, and the smallest unused number coprime to 2 that does not equal 2+1=3 is 5.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A064413, A093714, A121216, A347113.

nn = 76; c[_] = 0; a[1] = c[1] = 1; a[2] = c[2] = 2; u = 3
Do[k = u; While[Nand[c[k] == 0, CoprimeQ[#, k], k != # + 1], k++] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, May 01 2022 *)

A121217 a(1)=1, a(2)=2, a(3)=3; for n > 3, a(n) is the smallest positive integer which does not occur earlier in the sequence and which is not coprime to a(n-2).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 5, 14, 15, 7, 18, 21, 16, 24, 20, 22, 25, 11, 30, 33, 26, 27, 13, 36, 39, 28, 42, 32, 34, 38, 17, 19, 51, 57, 45, 48, 35, 40, 49, 44, 56, 46, 50, 23, 52, 69, 54, 60, 58, 55, 29, 65, 87, 70, 63, 62, 66, 31, 64, 93, 68, 72, 74, 75, 37, 78, 111, 76, 81

Offset: 1

Leroy Quet, Aug 20 2006

nonn

Conjecture: this is a permutation of the positive integers, cf. A256618. - Reinhard Zumkeller, Apr 05 2015

The B-sequence mentioned in the Maple program is not in the OEIS. It is 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, ... - Alois P. Heinz, Feb 02 2019

N. J. A. Sloane, Table of n, a(n) for n = 1..20632

Cf. A064413, A121216, A251622, A256414 (indices of primes), A256419 (smoothed version).

Cf. A256618 (conjectured inverse).

a121217 n = a121217_list !! (n-1)
a121217_list = 1 : 2 : 3 : f 2 3 [4..] where
   f u v xs = g xs where
     g (w:ws) = if gcd w u > 1 then w : f v w (delete w xs) else g ws
-- Reinhard Zumkeller, Apr 05 2015

# From N. J. A. Sloane, Apr 04 2015: A121217 gcd(A[n],A[n-2])>1 A=seq, for B see the COMMENTS
N:= 60: # to get a(1) to a(n) where a(n+1) is the first term > N
B:= Vector(N, datatype=integer[4]):
for n from 1 to 3 do A[n]:= n: od:
for n from 4 do
  for k from 4 to N do
    if B[k] = 0 and igcd(k, A[n-2]) > 1 then
       A[n]:= k;
       B[k]:= 1;
       break
    fi
  od:
  if k > N then break fi
od:
[seq(A[i], i=1..n-1)];

a = Range@ 3; Do[k = 4; While[Or[MemberQ[a, k], CoprimeQ[a[[i - 2]], k]], k++]; AppendTo[a, k], {i, 4, 72}]; a (* Michael De Vlieger, Aug 19 2017 *)

Extended by Ray Chandler, Aug 22 2006

A340783 a(n) = n if n <= 3; for n>3, a(n) = the closest number to a(n-1) that has not occurred earlier and has at least one common factor with a(n-2), but none with a(n-1). In case of a tie, choose the smaller.

Original entry on oeis.org

1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 24, 35, 34, 45, 44, 39, 38, 33, 32, 27, 26, 21, 20, 7, 10, 49, 48, 77, 76, 63, 62, 57, 56, 51, 50, 69, 68, 75, 74, 81, 80, 87, 86, 93, 92, 99, 98, 111, 110, 117, 116, 123, 122, 129, 128, 135, 134, 141, 140, 153, 152, 147, 146, 133, 132, 119, 118, 105, 104, 95

Offset: 1

Scott R. Shannon, Jan 21 2021

The sequence uses a similar selection rule to the Yellowstone permutation A098550 but instead of choosing the smallest number that has not occurred earlier that has a common factor with a(n-2) and no common factor with a(n-1), the number closest to a(n-1) that satisfies these rules is selected for a(n). If two such numbers are the same distance from a(n-1) then the smaller is chosen.
As any number n is coprime to n-1 there are many such pairs of values differing by one in the sequence. If a(n-1) is even and a(n) is odd then the only time a(n+1) will not be a(n)-1 is if a(n)-1 has already appeared in the sequence.
The majority of terms are clustered along a line with gradient approximately 1.25 . However the line is broken into smaller regions, each region having a slightly higher gradient, by the terms dropping to much smaller values before returning to the main line. See the linked image.
Excluding the first five terms, in the first 5 million terms the maximum number of consecutive terms that increase is only two. This only occurs at five places, when n=9,25,79,13705,275345. In the same range there are many regions of consecutive decreasing terms, the longest being 5997 terms starting from n=1902153.
In the first 5 million terms the only fixed points, other than the first three terms, are 4 and 313362. As the terms for larger n seem to drop below the a(n)=n line on numerous occasions it is possible more exist, although this is unknown. The smallest number not appearing is 31. It is unknown is all values eventually appear. In the same range the largest change in consecutive terms is from a(3503960)=30982 to a(3503961)=3191249, a difference of 3160267.

			a(5) = 9 as a(5-2) = a(3) = 3 so a(5) must have 3 as a factor, but cannot be 6 as it cannot have common factor with a(5-1) = a(4) = 2.
a(12) = 24 as a(12-2) = a(10) = 6 so a(12) must have 2 or 3 as a factor, but cannot have a factor with a(12-1) = a(11) = 25. The closest numbers to a(12-1) = a(11) = 25 which have not occurred and satisfy these criteria are 24 and 26, but 24 is chosen as it is the smaller of the two. This is the first term differing from A098550 as the later chooses the smallest number satisfying the criteria that has not occurred, namely 12.

Scott R. Shannon, Table of n, a(n) for n = 1..20000.
Scott R. Shannon, Image of the first 5 million terms. The green line is a(n)=n.

Cf. A098550, A336957, A064413, A121216.

A353708 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations.

Original entry on oeis.org

0, 1, 2, 4, 5, 3, 8, 12, 6, 16, 9, 7, 18, 24, 13, 32, 34, 10, 17, 20, 14, 11, 33, 36, 22, 19, 40, 44, 21, 64, 42, 15, 65, 48, 26, 66, 37, 25, 72, 38, 23, 73, 96, 50, 27, 68, 100, 35, 128, 28, 29, 67, 98, 52, 129, 74, 30, 49, 97, 70, 130, 41, 45, 80, 82, 39, 132, 88, 43, 131, 84, 56, 136, 69, 51, 58, 76, 133, 144, 90, 46, 160, 81, 31, 134, 192, 57, 47

Offset: 0

N. J. A. Sloane, May 06 2022

nonn

A set-theory analog of A121216.

This is a permutation of the nonnegative numbers.

Michael De Vlieger, Table of n, a(n) for n = 0..16384 (a(n) for n = 0..748 by N. J. A. Sloane)
Index entries for sequences that are permutations of the nonnegative integers

Cf. A109812, A121216.

read(transforms) : # ANDnos def'd here
A353708 := proc(n)
    option remember;
    local c, i, known ;
    if n <= 2 then
        n;
    else
        for c from 1 do
            known := false ;
            for i from 1 to n-1 do
                if procname(i) = c then
                    known := true;
                    break ;
                end if;
            end do:
            if not known and ANDnos(c, procname(n-2)) =0 then
                return c;
            end if;
        end do:
    end if;
end proc: # Following R. J. Mathar's program for A109812.
[seq(A353708(n), n=0..256)] ;
# second Maple program:
b:= proc() false end: t:= 2:
a:= proc(n) option remember; global t; local k; if n<2 then n
      else for k from t while b(k) or Bits[And](k, a(n-2))>0
      do od; b(k):=true; while b(t) do t:=t+1 od; k fi
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 06 2022

nn = 87; c[] = -1; a[0] = c[0] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == -1, BitAnd[a[n - 2], k] == 0], k++]; Set[{a[n], c[k]}, {k, n}]; If[k == u, While[c[u] > -1, u++]], {n, 2, nn}]; Array[a, nn + 1, 0] (* _Michael De Vlieger, May 06 2022 *)

A256399 First differences of A256219.

Original entry on oeis.org

1, 1, 3, 1, 3, 3, 1, 3, 8, 3, 5, 4, 3, 1, 4, 10, 1, 4, 4, 4, 4, 7, 4, 9, 3, 1, 4, 3, 4, 16, 4, 4, 4, 8, 1, 8, 7, 1, 8, 4, 4, 8, 3, 4, 4, 8, 16, 1, 3, 1, 11, 1, 4, 12, 7, 5, 3, 1, 8, 3, 5, 19, 4, 1, 3, 16, 5, 11, 1, 4, 4, 11, 5, 7

Offset: 1

N. J. A. Sloane, Mar 28 2015

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A121216, A256219.

cp's OEIS Frontend

A377091 a(0) = 0; thereafter a(n) is the least integer (in absolute value) not yet in the sequence such that the absolute difference between a(n-1) and a(n) is a square; in case of a tie, preference is given to the positive value.

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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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A353709 a(0)=0, a(1)=1; thereafter a(n) = smallest nonnegative integer not among the earlier terms of the sequence such that a(n) and a(n-2) have no common 1-bits in their binary representations and also a(n) and a(n-1) have no common 1-bits in their binary representations.

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A270139 a(n)=n when n<=3, otherwise a(n) is the smallest unused positive integer which is not coprime to the two previous terms.

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A352588 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that is coprime to a(n-1) but does not equal a(n-1)+1.

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A121217 a(1)=1, a(2)=2, a(3)=3; for n > 3, a(n) is the smallest positive integer which does not occur earlier in the sequence and which is not coprime to a(n-2).

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