A073666 -id:A073666 - cp's OEIS frontend

A092842 a(n) is the solution of the equation A073666(x) = n (A073666(a(n)) = n).

1, 2, 3, 4, 7, 6, 5, 8, 9, 12, 17, 10, 11, 14, 13, 21, 15, 16, 19, 23, 22, 20, 24, 30, 31, 25, 28, 29, 36, 18, 33, 35, 26, 27, 59, 34, 40, 37, 38, 41, 53, 32, 42, 50, 46, 43, 51, 39, 48, 45, 44, 47, 57, 49, 63, 52, 61, 62, 55, 56, 65, 54, 67, 74, 88, 58, 75, 60, 69, 64

Offset: 1

Farideh Firoozbakht, Apr 15 2004

nonn

For each n number of solutions of the equation A073666(x) =n is less than 2 because the terms of the sequence A073666 are distinct. Conjecture: This sequence is infinite(see A073666).

Cf. A073666, A081942, A081943, A092829.

A073667 Numbers which retain their position in A073666 (position not disturbed by the rearrangement).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 14, 19, 69, 71, 77, 150, 157, 231, 237, 445, 721, 725, 753, 819, 848, 875, 1402, 1445, 1467, 1476, 1485, 1492, 2381, 2722, 3003, 3116, 4528, 5132, 5157, 5329, 9302, 9429, 9537, 10014, 10568, 10694, 11278, 11482, 13186, 13816, 14306

Offset: 1

Amarnath Murthy, Aug 10 2002

nonn

			A073666(69) = 69, hence 69 is in the sequence.

Cf. A073666.

Corrected and extended by Klaus Brockhaus, Feb 13 2006

A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 10, 5, 15, 9, 18, 12, 16, 24, 30, 20, 40, 32, 48, 36, 27, 54, 72, 60, 45, 75, 25, 50, 70, 7, 14, 28, 42, 21, 63, 126, 84, 56, 112, 64, 96, 120, 80, 100, 150, 90, 108, 81, 162, 216, 144, 168, 140, 35, 105, 210, 180, 135, 225, 300

Offset: 1

Ivan Neretin, Apr 18 2015

Presumably a(n) is a permutation of the positive integers.
Primes seem to occur in their natural order. 31 appears as a(7060). Primes p >= 37 are not found among the first 10000 terms.
Numbers n such that a(n)=n are 1, 2, 3, 12, 306, ...
A256918(n) = gcd(a(n), a(n+1)); gcd(a(A257120(n)), a(A257120(n)+1)) = gcd(a(A257475(n)), a(A257475(n)-1)) = n. - Reinhard Zumkeller, Apr 25 2015
For p prime: A257122(p)-1 = index of the smallest multiple of p: a(A257122(p)-1) mod p = 0 and a(m) mod p > 0 for m < A257122(p)-1. - Reinhard Zumkeller, Apr 26 2015

			After a(9)=15, the values 1, 2, 3, 4, 6, and 8 are already used, while 7 is forbidden because gcd(15,7)=1 and that value of GCD has already occurred twice, at (1,2) and (2,3). The minimal value which is neither used not forbidden is 9, so a(10)=9.

Ivan Neretin and Reinhard Zumkeller, Table of n, a(n) for n = 1..25000, first 10000 terms from Ivan Neretin

Other minimal sequences of distinct positive integers that match some condition imposed on a(n) and a(n-1):
A175498 (differences are unique),
A081145 (absolute differences are unique),
A235262 (bitwise XORs are unique),
A163252 (differ by one bit in binary),
A000027 (GCD=1),
A064413 (GCD>1),
A128280 (sum is a prime),
A034175 (sum is a square),
A175428 (sum is a cube),
A077220 (sum is a triangular number),
A073666 (product plus 1 is a prime),
A081943 (product minus 1 is a prime),
A091569 (product plus 1 is a square),
A100208 (sum of squares is a prime).
Cf. A004526.
Cf. A256918, A257120, A257475, A257478, A257122 (putative inverse).
Cf. also A281978.

import Data.List (delete); import Data.List.Ordered (member)
a257218 n = a257218_list !! (n-1)
a257218_list = 1 : f 1 [2..] a004526_list where
   f x zs cds = g zs where
     g (y:ys) | cd `member` cds = y : f y (delete y zs) (delete cd cds)
              | otherwise       = g ys
              where cd = gcd x y
-- Reinhard Zumkeller, Apr 24 2015

a={1}; used=Array[0&,10000]; Do[i=1; While[MemberQ[a,i] || used[[l=GCD[a[[-1]],i]]]>=2, i++]; used[[l]]++; AppendTo[a,i], {n,2,100}]; a (* Ivan Neretin, Apr 18 2015 *)

A081943 a(1) = 1, a(n)= smallest number not occurring earlier such that a(n-1)*a(n) -1 is a prime. re-arrangement of natural numbers such that the product of adjacent terms is one more than a prime.

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 7, 12, 9, 8, 10, 11, 18, 13, 14, 16, 15, 24, 20, 19, 30, 17, 22, 21, 28, 23, 36, 27, 26, 33, 40, 32, 31, 42, 25, 50, 34, 38, 39, 48, 43, 60, 35, 46, 45, 44, 37, 54, 41, 52, 57, 56, 55, 68, 49, 62, 64, 53, 70, 29, 66, 63, 76, 47, 82, 69, 58, 51, 72, 61

Offset: 1

Amarnath Murthy, Apr 02 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..1000

Cf. A073666, A081942.

a[1]=1; a[n_]:=a[n]=(For[c=Sort[Table[a[k],{k,n-1}]]; d=Append[c,Last[c]+1]; m=First[Complement[Range[Last[d]],c]], MemberQ[c,m]||!PrimeQ[m*a[n-1]-1],m++ ]; m); Table[a[k],{k,70}] (* Farideh Firoozbakht, Apr 14 2004 *)

More terms from Farideh Firoozbakht, Apr 14 2004

A081942 a(1) = 1, a(n) = smallest number greater than a(n-1) such that a(n-1)*a(n) + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 13, 24, 25, 28, 34, 37, 40, 43, 46, 51, 56, 60, 67, 70, 79, 84, 87, 94, 105, 106, 120, 126, 130, 133, 136, 147, 148, 151, 156, 161, 162, 163, 166, 171, 176, 177, 184, 190, 193, 204, 208, 211, 228, 234, 239, 242, 248, 252, 256, 262, 265, 270

Offset: 1

Amarnath Murthy, Apr 02 2003

nonn

See A073666 for a nonincreasing version and A096100 for a more restrictive constraint. - M. F. Hasler, Nov 24 2015

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A081943, A073666, A096100.

f[s_List] := Block[{k = m = s[[-1]]}, k++; While[ !PrimeQ[k*m + 1], k++]; Append[s, k]]; Nest[f, {1}, 57] (* Robert G. Wilson v, Dec 02 2012 *)
smp[n_]:=Module[{m=n+1},While[!PrimeQ[m*n+1],m++];m]; NestList[smp,1,60] (* Harvey P. Dale, Dec 12 2018 *)

A081942(n,show=0,a=1)={for(n=2,n,show&&print1(a",");for(k=a+1,9e9, isprime(a*k+1) && (a=k) && break));a} \\ Use 2nd or 3rd optional arg to print intermediate terms or to use another starting value. - M. F. Hasler, Nov 24 2015

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Apr 08 2003

A308334 Lexicographically earliest sequence of distinct positive numbers such that for any n > 0, a(n) OR a(n+1) is a prime number (where OR denotes the bitwise OR operator).

Original entry on oeis.org

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

Offset: 1

Rémy Sigrist, May 20 2019

By Dirichlet's theorem on arithmetic progressions, we can always extend the sequence: say a(n) < 2^k, then a(n) OR 1 and 2^k are coprime and there are infinitely many prime numbers of the form (a(n) OR 1) + m*2^k = a(n) OR (1 + m*2^k) and we can extend the sequence.
Will every integer appear in this sequence?
Numerous sequences are based on the same model: the sequence is the lexicographically earliest sequence of distinct positive terms such that some function in two variables yields prime numbers when applied to consecutive terms:
  f(u,v)      Analog sequence
  -------     -----------------
  u OR v      a (this sequence)
  u + v       A055265
  u*v + 1     A073666
  u*v - 1     A081943
  abs(u-v)    A065186
  max(u,v)    A282649
  u^2 + v^2   A100208
The appearance of numbers much earlier or later than their corresponding index is flagged strikingly in the plot2 graph of a(n)/n (see links). - Peter Munn, Sep 10 2022

			The first terms, alongside a(n) OR a(n+1), are:
  n   a(n)  a(n) OR a(n+1)
  --  ----  --------------
   1     1               3
   2     2               3
   3     3               7
   4     4               5
   5     5               7
   6     6               7
   7     7              23
   8    16              29
   9    13              13
  10     8              11
  11    11              11
  12     9              11

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Peter Munn, Plot2 graph of a(n)/n

See A308340 for the corresponding prime numbers.

See A055265, A065186, A073666, A081943, A100208, A282649 for similar sequences.

s=0; v=1; for (n=1, 67, s+=2^v; print1 (v ", "); for (w=1, oo, if (!bittest(s,w) && isprime(o=bitor(v,w)), v=w; break)))

from sympy import isprime
from itertools import count, islice
def agen():
    aset, k, mink = {1}, 1, 2
    for n in count(1):
        an = k; yield an; aset.add(an)
        s, k = set(str(an)), mink
        while k in aset or not isprime(an|k): k += 1
        while mink in aset: mink += 1
print(list(islice(agen(), 67))) # Michael S. Branicky, Sep 10 2022

A092829 a(n) is the solution of the equation A081943(x) =n (a(A081943(n)) =n).

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 7, 10, 9, 11, 12, 8, 14, 15, 17, 16, 22, 13, 20, 19, 24, 23, 26, 18, 35, 29, 28, 25, 60, 21, 33, 32, 30, 37, 43, 27, 47, 38, 39, 31, 49, 34, 41, 46, 45, 44, 64, 40, 55, 36, 68, 50, 58, 48, 53, 52, 51, 67, 74, 42, 70, 56, 62, 57, 103, 61, 72, 54, 66, 59

Offset: 1

Farideh Firoozbakht, Apr 15 2004

nonn

For each n number of solutions of the equation A081943(x) =n is less than 2 because the terms of the sequence A081943 are distinct.

			a(25) =35 because A081943(35) =25.

Cf. A081943, A073666, A092842.

A282539 a(1)=1, a(n) is the smallest integer not included earlier such that a(n)a(n-1)-1 and a(n)a(n-1)+1 are twin primes.

Original entry on oeis.org

1, 4, 3, 2, 6, 5, 12, 9, 8, 24, 10, 15, 16, 27, 30, 14, 33, 20, 21, 22, 39, 28, 51, 42, 11, 18, 29, 72, 26, 57, 74, 45, 36, 23, 84, 13, 66, 7, 60, 17, 126, 38, 111, 68, 54, 37, 90, 31, 48, 44, 75, 34, 63, 56, 105, 32, 81, 50, 93, 106, 138, 40, 78, 19, 222, 110, 69, 58

Offset: 1

Alex Ratushnyak, Feb 17 2017

nonn

It is only a conjecture that a(n) always exists. - Editors, OEIS, Mar 07 2017

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

Cf. A000040, A001097, A014574, A073666.

from sympy import isprime
a = [1]
found = True
while found:
  found = False
  prev = a[len(a)-1]
  for k in range(2,10001):
    if isprime(prev*k-1) and isprime(prev*k+1) and (k not in a):
      a.append(k)
      found = True
      break
print(a)

cp's OEIS Frontend

A092842 a(n) is the solution of the equation A073666(x) = n (A073666(a(n)) = n).

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A073667 Numbers which retain their position in A073666 (position not disturbed by the rearrangement).

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A257218 Lexicographically earliest sequence of distinct positive integers such that gcd(a(n), a(n-1)) takes no value more than twice.

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A081943 a(1) = 1, a(n)= smallest number not occurring earlier such that a(n-1)*a(n) -1 is a prime. re-arrangement of natural numbers such that the product of adjacent terms is one more than a prime.

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A081942 a(1) = 1, a(n) = smallest number greater than a(n-1) such that a(n-1)*a(n) + 1 is prime.

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A308334 Lexicographically earliest sequence of distinct positive numbers such that for any n > 0, a(n) OR a(n+1) is a prime number (where OR denotes the bitwise OR operator).

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A092829 a(n) is the solution of the equation A081943(x) =n (a(A081943(n)) =n).

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A282539 a(1)=1, a(n) is the smallest integer not included earlier such that a(n)*a(n-1)-1 and a(n)*a(n-1)+1 are twin primes.

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A282539 a(1)=1, a(n) is the smallest integer not included earlier such that a(n)a(n-1)-1 and a(n)a(n-1)+1 are twin primes.