A382655 Lexicographically earliest sequence of distinct positive integers such that if m = a(n-1), a(n) is the smallest novel number k such that: g = gcd(m, k) > 1, m/g > 1, k/g >1 and gcd(m/g, k) = gcd(k/g, m) = 1.

6, 10, 14, 18, 22, 26, 30, 21, 12, 15, 20, 28, 35, 40, 24, 33, 39, 42, 34, 38, 46, 50, 54, 58, 62, 66, 51, 48, 57, 60, 44, 36, 45, 55, 65, 52, 68, 76, 84, 69, 75, 78, 70, 63, 56, 72, 88, 77, 91, 104, 117, 90, 74, 82, 86, 94, 98, 102, 85, 80, 95, 105, 87, 93, 96

Offset

1,1

Comments

The relation described in Name between two distinct numbers m, k refers to what may be described as "symmetric non-coprimality" (expressed as m # k), which in this sequence requires that if a(n-1) = m is a term, then a(n) = k is the smallest novel number such that m # k. From the definition it follows that 1, primes and prime powers cannot be terms in this sequence, however powerful numbers which are not prime powers can be terms (see Example). Sequence is conjectured to be a permutation of A024619. For two numbers m, k symmetrically related in this way we may write (m # r) = 1, and if the relation is not symmetric we may write (m # r) > 1. a(n) --> n as n-->00.

Links

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

Example

a(1) = 6 is the smallest number having the property that there exist numbers k such that 6 # k. Since 10 is the smallest such number, a(2) = 10.
a(3) cannot be 11 or 13 since they are prime, and cannot be 12 since with g = 2, 10/2 = 5 which is prime to 12, but 12/2 = 6 which is not prime to 10. Since gcd(10,14) = 2, 14/2 = 7 is prime to 10 and 10/2 = 5 is prime to 14 so a(3) = 14. a(4) = 18 is the first nonsquarefree term in the sequence. a(31) = 44 implies a(32) = 36 because g = 4 and gcd(44/4,36) = gcd(36/4,44) = 1 so 44 # 36, and 36 is the smallest number with this property. a(32) = 36 is the first powerful term in the sequence.

Mathematica

nn = 2^14; u = 2; c[_] := False; Set[{j, c[6]}, {6, True}];
fQ[x_, y_] := And[x != y, # > 1, x/# > 1, y/# > 1, GCD[x/#, y] == GCD[y/#, x] == 1] &[GCD[x, y]];
{j}~Join~Monitor[Reap[Do[k = u;
  While[Or[c[k], PrimePowerQ[k], ! fQ[j, k]], k++];
  c[k] = True; j = k; Sow[k];
If[k == u, While[Or[c[u], PrimePowerQ[u]], u++]], {n, nn}] ][[-1, 1]], n] (* Michael De Vlieger, Apr 02 2025 *)

Crossrefs

Cf. A001694, A013929, A024619.

Sequence in context: A115036 A332766 A315169 * A133653 A073760 A315170

Adjacent sequences: A382652 A382653 A382654 * A382656 A382657 A382658

Keyword

nonn

Author

David James Sycamore, Apr 01 2025

Status

approved