A353710 -id:A353710 - cp's OEIS frontend

A353718 Lengths of runs of identical terms in A353710.

1, 1, 1, 3, 8, 40, 3, 20, 10, 4, 95, 60, 77, 227, 498, 162, 438, 988, 334, 946, 1342, 13633, 1446, 810, 103, 140, 7033, 2518, 2369, 5096, 1719, 300, 2397, 14590, 434, 6539, 26193, 20403, 13857, 10, 26972, 24908, 44745, 3346, 149938, 5859, 29919, 132184, 123679

Offset: 1

N. J. A. Sloane, May 09 2022

			The first 60 terms of A353710 are 0, / 1, / 2, / 3, 3, 3, / 5, 5, 5, 5, 5, 5, 5, 5, / 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, / 11, 11, 11, / 15, 15, 15, ... The slashes indicate the initial runs of lengths 1, 1, 1, 3, 8, 40, 3, ...

Rémy Sigrist, Table of n, a(n) for n = 1..132
Rémy Sigrist, C++ program

Cf. A353709, A353710.

from itertools import count, islice
def A353718_gen(): # generator of terms
    s, a, b, c, ab, k = {0,1}, 0, 1, 2, 1, 1
    yield from (1,1)
    while True:
        for n in count(c):
            if not (n & ab or n in s):
                a, b = b, n
                ab = a|b
                s.add(n)
                if c in s:
                    yield k
                    k = 0
                    while c in s:
                        c += 1
                k += 1
                break
A353718_list = list(islice(A353718_gen(),20)) # Chai Wah Wu, May 10 2022

More terms from Rémy Sigrist, May 09 2022

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

A353723 Indices of records in A353717.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 15, 23, 27, 29, 31, 39, 47, 55, 61, 63, 95, 111, 123, 127, 191, 255, 431, 443, 447, 495, 511, 639, 703, 759, 763, 767, 879, 895, 943, 959, 1007, 1023, 1727, 1775, 1791, 1919, 2015, 2047, 2559, 3007, 3063, 3071, 3583, 3839, 3967, 4031, 4063, 4079, 4095, 6111, 6127, 6143, 7135, 7165, 7167, 7671, 7679, 7935, 8063, 8183, 8191, 11775, 11999, 12031

Offset: 1

Rémy Sigrist and N. J. A. Sloane, May 10 2022

nonn

These are the numbers that are the slowest to appear in A353709.

Also distinct values of A353710; a(n) appears A353718(n) times in A353710.

Rémy Sigrist, Table of n, a(n) for n = 1..132
Rémy Sigrist, C++ program

Cf. A353709, A353710, A353717, A353718, A353720, A353721, A353722.

cp's OEIS Frontend

A353718 Lengths of runs of identical terms in A353710.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Python

A353723 Indices of records in A353717.

Views

Author

Keywords

Comments

Links

Crossrefs