A355061 Lexicographically earliest infinite sequence of positive numbers such that, for n>2, a(n) has a common factor with a(n-1), no common factor with a(n-2), and the product a(n)*a(n-1) is distinct from all previous products, a(i)*a(i-1), i=2..n-1.

1, 2, 6, 15, 35, 14, 6, 33, 55, 10, 6, 21, 35, 10, 12, 21, 77, 22, 6, 39, 65, 10, 14, 21, 15, 10, 22, 33, 15, 20, 14, 63, 15, 40, 14, 77, 33, 12, 14, 91, 39, 12, 20, 35, 63, 6, 26, 65, 15, 12, 22, 55, 15, 18, 28, 35, 45, 12, 26, 91, 21, 30, 22, 143, 39, 15, 50, 22, 99, 15, 70, 22, 187, 51, 6

Offset

1,2

Comments

Like the Enots Wolley sequence, A336957, no term a(n) can be a prime or a prime power as this would make it impossible to find a(n+1). As 6 is the smallest number to include two different primes, and hence the smallest number beyond the first two terms that can appear, it occurs frequently in the sequence, 1887 times in the first 250000 terms. See A355139 for the indices of these terms.

Unlike A336957 multiple odd successive terms occur, the longest such run in the first 250000 terms being fourteen starting at a(111799) = 20257.

See A355138 for the products of consecutive terms.

Links

Table of n, a(n) for n=1..75.

Scott R. Shannon, Image of the first 250000 terms. The green line is y = n.

Example

a(5) = 35 as this is the smallest number to share a factor with a(4) = 15, not share a factor with a(3) = 6, and contains a prime factor not in a(4) = 15 and hence allows a(6) to exist.
a(7) = 6 as this is the smallest number to share a factor with a(6) = 14, not share a factor with a(5) = 35, and contains a prime factor not in a(6) = 14 and hence allows a(8) to exist. This is the first term to differ from A336957.

Prog

(Python)
from sympy import primefactors
from itertools import count, islice
def agen(): # generator of terms
    an1, an, f1, f, pset = 2, 6, {2}, {2, 3}, {2, 12}
    yield from [1, 2, 6]
    for n in count(4):
        an2, an1, an, f2, f1 = an1, an, 6, f1, f
        f = set(primefactors(an))
        while an*an1 in pset or f1&f == set() or f2&f != set() or f <= f1:
            an += 1; f = set(primefactors(an))
        pset.add(an*an1); yield an
print(list(islice(agen(), 75))) # Michael S. Branicky, Jun 20 2022

Crossrefs

Cf. A355138, A355139, A336957, A098550, A064413.

Sequence in context: A221719 A095380 A287012 * A336957 A338055 A336799

Adjacent sequences: A355058 A355059 A355060 * A355062 A355063 A355064

Keyword

nonn

Author

Scott R. Shannon, Jun 16 2022

Status

approved