A373623 -id:A373623 - cp's OEIS frontend

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

Scott R. Shannon, Jun 13 2024

nonn

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.

			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.

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.

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

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 *)

A373860 a(n) = n for n <= 4; for n > 4, a(n) is the smallest unused positive number that shares a factor with the most recently appearing even number prior to a(n-1) if a(n-1) is even, otherwise it shares a factor with the most recently appearing odd number prior to a(n-1) if a(n-1) is odd.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 10, 14, 5, 15, 20, 7, 18, 16, 21, 28, 22, 24, 11, 27, 33, 30, 26, 25, 36, 13, 35, 39, 40, 32, 34, 38, 17, 42, 19, 51, 57, 45, 48, 44, 46, 50, 23, 54, 52, 56, 58, 49, 69, 63, 60, 29, 66, 55, 87, 65, 72, 62, 64, 31, 70, 68, 74, 76, 37, 93, 111, 75, 78, 80, 81, 84, 82, 77, 90

Offset: 1

Scott R. Shannon, Jun 19 2024

nonn

The terms appear to follow a pattern similar to the EKG sequence A064413, i.e., the terms are concentrated along just three lines of different gradients, and the lower line consists only of primes. The uppermost line appears to be composed only of semiprimes that are divisible by 3, while the middle line contains all other terms. See the attached images. For the first 100000 terms the primes appear in their natural order, implying that is likely true for all n.

The fixed points are 1, 2, 3, 4, 16, 32, 124, and it is likely that no more exist. The sequence is conjectured to be a permutation of the positive numbers.

			a(8) = 12 as a(7) = 9 is odd, and the most recently appearing odd number prior to a(7) is a(3) = 3, and 12 is the smallest unused positive number to share a factor with 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 5000 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.

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

c[] := False; j = 4; nn = 120; q[] := 0; m[_] := 1;
Array[Set[{a[#], c[#]}, {#, True}] &, j]; q[0] = 4; q[1] = 3; u = 5;
Do[If[EvenQ[j],
  If[PrimePowerQ[q[0]], k = m[2];
   While[c[2 k], k++]; k *= 2; While[c[2 m[2]], m[2]++],
   k = u; While[Or[c[k], CoprimeQ[q[0], k]], k++]],
  If[PrimePowerQ[q[1]],
   (k = m[#]; While[c[k #], k++]; k *= #;
      While[c[#  m[#]], m[#]++]) &[FactorInteger[q[1]][[1, 1]]],
   k = u; While[Or[c[k], CoprimeQ[q[1], k]], k++]]
  ]; q[Mod[j, 2]] = j;
 Set[{a[n], c[k], j}, {k, True, k}];
 If[k == u, While[c[u], u++]], {n, j + 1, nn}];
Array[a, nn] (* Michael De Vlieger, Jun 20 2024 *)

from math import gcd, lcm
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 2, 3, 4]
    aset, an, eo, mink = {1, 2, 3, 4}, 4, [2, 3], 5
    while True:
        s, prevan = eo[an%2], an
        an = next(k for k in count(mink) if k not in aset and gcd(s,k)>1)
        eo[prevan%2] = prevan
        aset.add(an)
        while mink in aset: mink += 1
        yield an
print(list(islice(agen(), 77))) # Michael S. Branicky, Jun 20 2024

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica