A308653 Lexicographically earliest sequence of distinct composite numbers such that for any n > 0, gcd(a(n), a(n+1)) > lpf(a(n)) (where lpf = A020639).

4, 8, 12, 6, 9, 18, 15, 10, 20, 16, 24, 21, 14, 28, 32, 36, 27, 45, 25, 50, 30, 33, 22, 44, 40, 35, 42, 39, 26, 52, 48, 51, 34, 68, 56, 49, 98, 63, 54, 57, 38, 76, 60, 55, 66, 69, 46, 92, 64, 72, 75, 65, 78, 81, 90, 70, 77, 88, 80, 84, 87, 58

Offset

1,1

Comments

A more explicit (but longer) Name is: a(1) = 4. For n > 1, a(n) is the composite determined as follows: exclude smallest divisor > 1 of a(n-1), and of the remaining divisors > 1 select one such that a(n) is the smallest composite number not yet in the sequence that has this divisor in common with a(n-1).

By definition of the sequence, the divisors of a(n-1) used to determine a(n) are either primes or powers of primes.

Links

Table of n, a(n) for n=1..62.

Example

The smallest divisor (greater than 1) of a(1) = 4 is 2, hence an allowed divisor is 4 and a(2) = 8.
The smallest divisor (greater than 1) of a(3) = 12 is 2, hence allowed divisors are 4 and 3.  If 4 is chosen a(4) = 16, and if 3 is chosen, a(4) = 6.  Hence 3 is chosen and a(4) = 6.
Of the allowed divisors of a(n-1), not always the smallest one is chosen to determine a(n).  For example, a(25) = 40, so the smallest divisor greater than 1 is 2, and allowed divisors are 4 and 5.  If 4 is chosen, a(26) = 48, and if 5 is chosen, a(26) = 35, so 5 is chosen and a(26) = 35.

Crossrefs

Cf. A002808.

Sequence in context: A103696 A196267 A004469 * A196268 A251756 A357803

Adjacent sequences: A308650 A308651 A308652 * A308654 A308655 A308656

Keyword

nonn

Author

Enrique Navarrete, Sep 30 2019

Extensions

Name edited by Felix Fröhlich, Oct 09 2019

New definition from Rémy Sigrist, Oct 19 2019

Status

approved