A373998 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 5; for n > 4, a(n) is the smallest unused positive number that is coprime to a(n-1) and a(n-2) but has a common factor with at least one of a(1)...a(n-3).

1, 2, 3, 5, 4, 9, 25, 8, 21, 55, 16, 7, 11, 6, 35, 121, 12, 49, 65, 18, 77, 13, 10, 27, 91, 20, 33, 119, 26, 15, 17, 14, 39, 85, 22, 57, 115, 28, 19, 23, 24, 95, 143, 32, 45, 133, 34, 69, 125, 38, 51, 145, 44, 63, 29, 40, 81, 161, 50, 87, 169, 46, 75, 187, 52, 93, 175, 58, 31, 99, 56, 155

Offset: 1

Author

Scott R. Shannon, Jun 24 2024

Keywords

• nonn

Comments

For the terms studied, and similar to A373390 and A089088, the terms are concentrated along distinct lines of different gradient, four in this sequence, and like those sequences, the lowermost line is composed only of primes.

Note that in A089088, A373390, and this sequence, a(n) is coprime to zero, one and two previous terms, and the corresponding sequence terms are concentrated along two, three and four lines respectively.

The fixed points are 1, 2, 3, 8, 45, 51, and it is likely no more exist. For the terms studied the primes appear in their natural order. The sequence is conjectured to be a permutation of the positive integers.

Examples

			a(7) = 25 as 25 is the smallest unused number that is coprime to a(5) = 4 and a(6) = 9 while sharing a factor with a(4) = 5.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 50000 terms. Numbers with one, two, three, four, or five and more prime factors, counted with multiplicity, are show as red, yellow, green, blue and violet respectively.

Crossrefs

Cf. A373999, A373390, A089088, A064413, A098550, A373880.

Programs

Python

from math import gcd, lcm
from itertools import count, islice
def agen(): # generator of terms
    alst = [1, 2, 3, 5]
    yield from alst
    aset, LCM, mink = set(alst), lcm(*alst[:-2]), 4
    while True:
        an = next(k for k in count(mink) if k not in aset and 1 == gcd(k, alst[-1]) == gcd(k, alst[-2]) and gcd(k, LCM) > 1)
        LCM = lcm(LCM, alst[-2])
        alst.append(an)
        aset.add(an)
        while mink in aset: mink += 1
        yield an
print(list(islice(agen(), 72))) # Michael S. Branicky, Jun 24 2024