A103683 -id:A103683 - cp's OEIS frontend

A105214 Companion sequence to A103683: a(n) = k if A103683(k) = n; a(n) = 0 if n is not in {A103683}.

1, 2, 3, 6, 4, 46, 5, 10, 7, 22, 8, 50, 9, 14, 11, 18, 12, 54, 13, 26, 21, 30, 15, 58, 16, 34, 17, 42, 19, 62, 20, 38, 25, 70, 31, 66, 23, 74, 29, 78, 24, 102982, 27, 86, 41, 82, 28, 102974, 35, 98, 33, 90, 32, 102978, 36, 102, 37, 106, 39, 102990, 40, 94, 71

Offset: 1

Leroy Quet and Robert G. Wilson v, Apr 13 2005

nonn

A103683 is injective.
Inverse of A103683.
Conjecture: a(42 + 6*k) = 0.  This would follow from the conjecture that for n >= 67, A103683(n) is divisible by 3 iff n == 3 mod 4 and by 2 iff n == 2 mod 4. - Robert Israel, May 12 2015
The conjecture is false; a(42) = 102982. - Rémy Sigrist, Jan 05 2022

			a(6) = 46 because A103683(46) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A105214

Cf. A103683.

C
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See Links section.
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f[s_] := Block[{k = 1, l = Take[s, -3]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Take[Ordering@ Nest[f, {1, 2, 3}, 200], 41] (* Robert G. Wilson v, Jun 26 2011 *)

More terms from Rémy Sigrist, Jan 05 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

A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

Original entry on oeis.org

1, 3, 5, 7, 2, 9, 11, 13, 4, 15, 17, 19, 8, 21, 23, 25, 16, 27, 29, 31, 10, 33, 37, 41, 14, 39, 43, 47, 20, 49, 51, 53, 22, 35, 57, 59, 26, 55, 61, 63, 32, 65, 67, 69, 28, 71, 73, 45, 34, 77, 79, 75, 38, 83, 89, 81, 40, 91, 97, 87, 44, 85, 101, 93, 46, 95, 103, 99, 52, 107, 109, 105

Offset: 1

David James Sycamore, Apr 10 2022

nonn

The pairwise coprime relations found in the first four odd numbers 1,3,5,7 are preserved throughout in any run of four consecutive terms.
a(4n+5) is always even (and < a(4n+2)); n>=0.
The plot exhibits two distinct rays at first (upper/odd, lower/even), with no terms divisible by 6 until a(229), at which point the even ray switches to producing just 28 multiples of 6 until a(337)=168. At this point the original even ray is re-established, the odd ray divides into two (quasi-parallel) rays, and no further multiples of 6 are seen. Therefore it seems very unlikely that the sequence is a permutation of the nonnegative integers.
Primes p other than p = 2 appear in their natural order.

			3,5,7 are pairwise coprime and 2 is the smallest unused number coprime to all of them, therefore a(5)=2.

Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, highlighting primes in green, numbers divisible by 6 in gold, other even numbers in red, odd numbers divisible by 3 in blue.

Cf. A085229, A103683, A350359.

ina := proc(n) false end: # adapted from code for A103683
a := proc (n) option remember; local k;
if n < 5 then k := 2*n-1
else for k from 2 while ina(k) or igcd(k, a(n-1)) <> 1 or igcd(k, a(n-2)) <> 1 or igcd(k, a(n-3)) <> 1
do
end do
end if; ina(k):= true; k
end proc;
seq(a(n), n = 1 .. 100);

Block[{a, c, k, m, u, nn}, nn = 86; c[] = 0; MapIndexed[Set[{a[First[#2]], c[#1]}, {#1, First[#2]}] &, {1, 3, 5, 7}]; u = 2; Do[k = u; m = LCM @@ Array[a[i - #] &, 3]; While[Nand[c[k] == 0, CoprimeQ[m, k]], k++]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 5, nn}]; Array[a, nn] ] (* _Michael De Vlieger, Apr 12 2022 *)

from math import gcd
from itertools import islice
def agen(): # generator of terms
    aset, b, c, d = {1, 3, 5, 7}, 3, 5, 7
    yield from [1, b, c, d]
    while True:
        k = 1
        while k in aset or any(gcd(t, k) != 1 for t in [b, c, d]): k+= 1
        b, c, d = c, d, k
        aset.add(k)
        yield k
print(list(islice(agen(), 107))) # Michael S. Branicky, Apr 10 2022

A143345 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms (or n-1 terms if n<5) and does not occur earlier.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 4, 9, 13, 17, 19, 8, 15, 23, 29, 31, 14, 25, 27, 37, 41, 16, 35, 33, 43, 47, 26, 49, 45, 53, 59, 22, 61, 21, 65, 67, 32, 71, 51, 55, 73, 28, 79, 39, 83, 85, 38, 77, 69, 89, 97, 10, 91, 57, 101, 103, 20, 107, 63, 109, 113, 34, 95, 81, 121, 127, 46, 119, 75, 131, 137, 44, 133, 87, 115, 139, 52, 149, 93, 125, 151, 56, 143, 111, 145, 157, 62, 161, 99, 163, 167, 40, 169, 123, 173, 179, 50, 181

Offset: 1

M. F. Hasler, Jan 18 2011

nonn

One possible extension of A084937, A103683 to N=4. Here, a(4)=5 is chosen such that a(n) is coprime to a(k) for 0 < k < n <= 4. Another choice is a(k)=k (k<=4), which yieds the different sequence A180348.
It appears that:
- no multiples of 6 occur in this sequence, so it is not a permutation of the integers.
- a(n)=3 (mod 6)  iff  n=3, n=8, n=13 or n=14+5k, k>0.
- a(n)=0 (mod 2) iff n= 2+5k, k>=0.
- powers of 2 occur in natural order.
- powers of 3 occur in natural order.
- powers of any prime p occur in natural order.
- powers of any number occur in natural order.

Zak Seidov, Table of n, a(n) for n = 1..5000

print1("1,2,3"); a=[1,2,3,L=5]; unused=[4]; v=vector(#a,i,1); for(n=4,99, print1(","a[#a]); for(i=1,#unused,apply(x->gcd(x,unused[i]),a)==v | next; a=concat(vecextract(a,"^1"),unused[i]);unused=vecextract(unused,Str("^",i));next(2));L++;while(apply(x->gcd(x,L),a) !=v,unused=concat(unused,L++-1););a=concat(vecextract(a,"^1"),L))

A180348 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms and does not occur earlier; a(n) = n for n=1,2,3,4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 9, 8, 13, 17, 19, 15, 14, 23, 29, 31, 25, 6, 37, 41, 43, 35, 12, 47, 53, 59, 49, 10, 27, 61, 67, 71, 16, 21, 55, 73, 79, 26, 51, 77, 83, 89, 20, 39, 97, 101, 103, 22, 45, 91, 107, 109, 32, 33, 65, 113, 119, 38, 69, 121, 125, 127, 28, 57, 131, 85, 137, 44, 63, 139, 95, 149, 34, 81, 143, 115, 133, 58

Offset: 1

M. F. Hasler, Jan 18 2011

nonn

One possible generalization of A084937 resp. A103683 to N=4, cf. A143345 for a different version.

N=99; print1("1, 2, 3"); a=[1, 2, 3, L=4]; unused=[]; v=vector(#a, i, 1); for(n=#a, N, print1(", "a[#a]); for(i=1, #unused, apply(x->gcd(x, unused[i]), a)==v || next; a=concat(vecextract(a, "^1"), unused[i]); unused=vecextract(unused, Str("^", i)); next(2)); L++; while(apply(x->gcd(x, L), a) !=v, unused=concat(unused, L++-1); ); a=concat(vecextract(a, "^1"), L))

A291588 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and k >= 0, gcd(a(n), a(n + 2^k)) = 1.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 17, 8, 19, 9, 10, 23, 29, 14, 27, 25, 16, 31, 37, 12, 35, 41, 22, 43, 39, 20, 47, 49, 32, 33, 53, 26, 59, 61, 15, 67, 71, 28, 73, 45, 34, 79, 77, 38, 65, 83, 46, 89, 21, 40, 97, 91, 44, 51, 95, 58, 101, 103, 18, 55, 107, 52, 109

Offset: 1

Rémy Sigrist, Aug 27 2017

nonn

For a nonempty subset of the natural numbers, say S, let f_S be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0 and s in S, gcd(a(n), a(n + s)) = 1:
- f_S is well defined (we can always extend the sequence with a new prime number),
- f_S(1) = 1, f_S(2) = 2, f_S(3) = 3,
- all prime numbers appear in f_S, in increasing order,
- if a(k) = p for some prime p, then k <= p and max_{i=1..k} a(i) = p,
- in particular:
  S               f_S
  ---------       ---
  { 1 }           A000027 (the natural numbers)
  { 2 }           A121216
  { 1, 2 }        A084937
  { 1, 2, 3 }     A103683
  { 1, 2, 3, 4 }  A143345
  A000027         A008578 (1 alongside the prime numbers)
  A000079         a (this sequence)
- see also Links section for the scatterplots of f_S for certain classical S sets,
- likely f_S = f_S' iff S = S'.
The motivation for this sequence is to have a sequence f_S for some infinite subset S of the natural numbers.

			a(1) = 1 is suitable.
a(2) must be coprime to a(2 - 2^0) = 1.
a(2) = 2 is suitable.
a(3) must be coprime to a(3 - 2^0) = 2, a(3 - 2^1) = 1.
a(3) = 3 is suitable.
a(4) must be coprime to a(4 - 2^0) = 3, a(4 - 2^1) = 2.
a(4) = 5 is suitable.
a(5) must be coprime to a(5 - 2^0) = 5, a(5 - 2^1) = 3, a(5 - 2^2) = 1.
a(5) = 4 is suitable.
a(6) must be coprime to a(6 - 2^0) = 4, a(6 - 2^1) = 5, a(6 - 2^2) = 2.
a(6) = 7 is suitable.
a(7) must be coprime to a(7 - 2^0) = 7, a(7 - 2^1) = 4, a(7 - 2^2) = 3.
a(7) = 11 is suitable.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000040 (the prime numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000045 (the Fibonacci numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000142 (the factorial numbers)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000244 (the powers of 3)
Rémy Sigrist, Scatterplot of the first 5000 terms of f_A000312 (A000312(k) = k^k)
Rémy Sigrist, PARI program for A291588
Rémy Sigrist, Scatterplot of the first 500000 terms

Cf. A000027, A000040, A000045, A000079, A000142, A000244, A000312, A008578, A084937, A103683, A121216, A143345.

PARI
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\\ See Links section.
```

A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 9, 8, 11, 13, 6, 17, 19, 10, 21, 23, 16, 29, 27, 20, 31, 37, 12, 41, 43, 14, 15, 47, 22, 53, 39, 32, 25, 49, 18, 59, 61, 34, 45, 67, 38, 71, 33, 28, 65, 73, 24, 79, 83, 46, 75, 89, 56, 97, 81, 44, 85, 101, 26, 87, 103, 62, 35, 57, 64, 107

Offset: 1

Sergio Pimentel, May 14 2021

nonn

It is not clear if every positive integer is in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A344307

Cf. A064413, A084937, A090252, A103683, A121216, A249064.

PARI
```
See Links section.
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More terms from Rémy Sigrist, May 29 2021

A115928 Smallest number which is coprime to the last four predecessors and not occurring earlier; a(1)=1, a(2)=2, a(3)=3 & a(4)=4.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 13, 15, 14, 17, 19, 23, 6, 25, 29, 31, 12, 35, 37, 41, 16, 21, 43, 47, 10, 27, 49, 53, 20, 33, 59, 61, 26, 45, 67, 71, 22, 39, 73, 79, 28, 51, 55, 83, 32, 57, 65, 77, 34, 69, 89, 91, 38, 75, 97, 101, 44, 63, 85, 103, 46, 81, 195, 107, 52, 87, 109, 113, 40, 93, 119

Offset: 1

Robert G. Wilson v, Jun 26 2011

Cf. A084937, A103683.

f[s_] := Block[{k = 1, l = Take[s,-4]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s,k], k++]; Append[s,k]]; Nest[f, {1,2,3,4}, 70]

cp's OEIS Frontend

A105214 Companion sequence to A103683: a(n) = k if A103683(k) = n; a(n) = 0 if n is not in {A103683}.

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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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A352950 Lexicographically earliest infinite sequence of distinct nonnegative integers commencing 1,3,5,7 such that any four consecutive terms are pairwise coprime.

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A143345 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms (or n-1 terms if n<5) and does not occur earlier.

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A180348 Lexicographically earliest sequence such that a(n) is coprime to the preceding 4 terms and does not occur earlier; a(n) = n for n=1,2,3,4.

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A291588 Lexicographically earliest sequence of distinct positive terms such that for any n > 0 and k >= 0, gcd(a(n), a(n + 2^k)) = 1.

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A344307 a(n) is the smallest number not yet in the sequence that satisfies the following condition: if p is a prime factor of a(n), the next p terms are coprime to p.

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A115928 Smallest number which is coprime to the last four predecessors and not occurring earlier; a(1)=1, a(2)=2, a(3)=3 & a(4)=4.

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