A328278 a(1) = 4. For n > 1, a(n) is the smallest composite number not yet in the sequence that has a divisor in common with a(n-1) other than the largest proper divisor of a(n-1).

4, 8, 6, 10, 12, 9, 18, 14, 16, 20, 15, 21, 24, 22, 26, 28, 30, 25, 50, 32, 34, 36, 27, 33, 39, 42, 35, 40, 38, 44, 46, 48, 45, 51, 54, 52, 56, 49, 98, 58, 60, 55, 65, 70, 62, 64, 66, 57, 63, 69, 72, 68, 74, 76, 78, 75, 80, 82, 84, 77, 91

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 the largest proper divisor of a(n-1), and of the remaining divisors > 1 of a(n-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..61.

Example

a(2) = 8 since after excluding 2, which is the largest proper divisor of a(1) = 4, the only remaining divisor > 1 of 4 is 4, and a(2) = 8 is the smallest composite not yet in the sequence that shares this divisor with a(1) = 4.
a(6) = 9 since after excluding 6, which is the largest proper divisor of a(5) = 12, among the remaining divisors > 1 of 12 are 2 and 3; if 2 is selected, a(6) = 14, and if 3 is selected, a(6) = 9, so 3 is selected and a(6) = 9.
a(7) = 18 since after excluding 3, which the largest proper divisor of a(6) = 9, the only remaining divisor > 1 of 9 is 9, and a(7) = 18 is the smallest composite not yet in the sequence that shares this divisor with a(6) = 9.
Note that after excluding the largest proper divisor of a(n-1), not always the smallest divisor > 1 of a(n-1) is selected to determine a(n), as a(6) shows.

Crossrefs

Cf. A002808, A308653.

Sequence in context: A142350 A011515 A005531 * A288189 A335159 A064494

Adjacent sequences: A328275 A328276 A328277 * A328279 A328280 A328281

Keyword

nonn

Author

Enrique Navarrete, Oct 10 2019

Status

approved