A384503 Lexicographically earliest infinite sequence of distinct positive integers having the property that for any pair a(n-2) = i, a(n-1) = j of consecutive terms > 1, a(n) is the smallest novel k such that gcd(i,k) > 1 if gcd(i,j) = 1 or gcd(j,k) = 1 if gcd(i,j) > 1.

1, 2, 3, 4, 6, 5, 8, 10, 7, 12, 14, 9, 16, 15, 18, 11, 20, 22, 13, 24, 26, 17, 28, 34, 19, 30, 38, 21, 32, 27, 36, 23, 33, 46, 39, 40, 42, 25, 35, 29, 45, 58, 48, 31, 44, 62, 37, 50, 74, 41, 52, 82, 43, 54, 86, 47, 56, 94, 49, 60, 63, 53, 51, 106, 57, 64, 66, 59

Offset

1,2

Comments

Three initial terms (1,2,3) are needed since starting 1,2 would require 1 to have a prime factor.

Similar to A280985 except that if for some m, a(m) = prime p, then a(m+2) = 2*p (whereas A280985(m+1) = 2*p).

Conjectured to be a permutation of the positive integers, with primes occurring in order.

Links

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

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^10, showing primes in red, prime powers in gold, squarefree composites in green, and numbers neuther squarefree nor prime powers in blue and magenta, with magenta additionally representing powerful numbers that are not prime powers.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Formula

For prime a(n) = p, a(n+2) = 2*p. - Michael De Vlieger, May 31 2025

Example

The lexicographically earliest condition requires that the sequence starts a(n) = n for n <= 3. Then with a(2) = 2 and a(3) = 3 a(4) must be 4, the smallest novel number sharing a prime divisor with 2 (since gcd(2,3) = 1). Since gcd(3,4) = 1, a(5) must be 6, the smallest novel number sharing a prime divisor with 3. Since gcd(4,6) > 1 a(6) = 5, the smallest novel number prime to a(4) = 4. a(8) = 8 because gcd(6,5) = 1 and then a(9) = 2*5 = 10 the smallest novel number sharing a factor with 5.

Mathematica

nn = 120; c[_] := False; m[_] := 1; c[1] = c[2] = c[3] = True; i = 2; j = 3; u = 4;
Range[3]~Join~Reap[Do[
  If[CoprimeQ[i, j],
    If[PrimePowerQ[i],
      p = FactorInteger[i][[1, 1]]; While[c[p*m[p]], m[p]++]; k = p*m[p],
      k = u; While[Or[c[k], CoprimeQ[i, k]], k++] ],
    k = u; While[Or[c[k], ! CoprimeQ[j, k]], k++] ];
  Sow[k]; Set[{c[k], i, j}, {True, j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}] ][[-1, 1]] (* Michael De Vlieger, May 31 2025 *)

Crossrefs

Cf. A000027, A064413, A280985, A127202.

Sequence in context: A379746 A095424 A194507 * A118316 A197756 A080738

Adjacent sequences: A384500 A384501 A384502 * A384504 A384505 A384506

Keyword

nonn

Author

David James Sycamore, May 31 2025

Status

approved