A373698 a(n) = n for n <= 3; for n > 3, a(n) is the smallest unused positive number that shares a factor with the largest odd divisor of a(n-1).

1, 2, 3, 6, 9, 12, 15, 5, 10, 20, 25, 30, 18, 21, 7, 14, 28, 35, 40, 45, 24, 27, 33, 11, 22, 44, 55, 50, 60, 36, 39, 13, 26, 52, 65, 70, 42, 48, 51, 17, 34, 68, 85, 75, 54, 57, 19, 38, 76, 95, 80, 90, 63, 49, 56, 77, 66, 69, 23, 46, 92, 115, 100, 105, 72, 78, 81, 84, 87, 29, 58, 116, 145, 110

Offset: 1

Author

Scott R. Shannon, Jun 13 2024

Keywords

• nonn

Comments

The terms studied are all concentrated along lines of different gradient, the prime numbers, which appear in their natural order, forming the lowermost line. The central line has the highest concentration of terms and is composed of numbers with various numbers of prime factors, while many of the other lines consist of terms with the same number of prime factors. The odd terms all fall into the highest, lowest, and middle line, all other lines containing only even terms. It is conjectured the above behaviors are true for all n. See the attached images.

It appears that when a prime p > 3 is a term, it is immediately preceded by a term 3*p and followed by a term 2*p. This is true for at least the first 500000 terms. This pattern is similar to that seen in the EKG sequence A064413 but here the order of appearance of 2*p and 3*p is reversed.

Other than the first three terms the first fixed points are 42544 and 47312, and no more exist in the first 500000 terms. An examination of the graphs shows these points are due to one of the lines of concentration crossing the line a(n) = n, implying it, and probably most of the other lines, has a slight downward curvature.

In the first 100000 terms the longest run of consecutive odd and even terms is eight, starting n = 96230 and n = 30466 respectively. It in unknown if these runs can be of arbitrary length. No power of 2, other than 2 itself, can be a term.

Examples

			a(11) = 25 as a(10) = 20 which has a largest odd divisor of 5, and 25 is the smallest unused positive number that shares a factor with 5.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, perfect powers of primes in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and purple, where the latter additionally represents powerful numbers that are not prime powers.
• Scott R. Shannon, Image of the first 100000 terms. Numbers with one, two, three, four, or five and more prime factors, counted with multiplicity, are shown as red, yellow, green, blue and violet respectively. The white line is a(n) = n.
• Scott R. Shannon, Image of the first 100000 terms. The even terms are shown in yellow, the odd terms in green.

Crossrefs

Cf. A373623, A373545, A373546, A064413, A005843, A005408.

Programs

Mathematica

j = 3; nn = 120; c[] := False; m[] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, j]; u = j + 1;
Do[If[PrimePowerQ[j],
    (k = m[#]; While[c[k #], k++]; k *= #; While[c[#  m[#]], m[#]++]) &@
      FactorInteger[j][[1, 1]],
     k = u; While[Or[c[k], CoprimeQ[j, k]], k++] ];
  Set[{a[n], c[k], j}, {#, True, #/2^IntegerExponent[#, 2]}] &[k],
  {n, j + 1, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 20 2024 *)