A338055 Lexicographically earliest infinite sequence {a(n)} of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1) involving only primes <= 11 but no such common factor with a(n-2) (primes > 11 play no role in this definition).

1, 2, 6, 15, 35, 14, 12, 33, 55, 10, 18, 21, 77, 22, 20, 45, 63, 28, 40, 75, 99, 44, 50, 105, 231, 88, 80, 135, 147, 56, 100, 165, 189, 98, 110, 225, 441, 112, 160, 275, 297, 24, 70, 385, 363, 36, 140, 539, 891, 30, 175, 847, 66, 60, 245, 3773, 132, 90, 875, 5929, 176, 48

Offset: 1

Author

N. J. A. Sloane, Oct 11 2020, based on an email from Frank Stevenson, Aug 26 2020

Keywords

• nonn

Comments

Let p_i denote the i-th prime. If the prime decompositions of x and y are

x = Product_{i=1..5} p_i^e_i*q_x, y = Product_{i=1..5} p_i^f_i*q_y,

then we define gcd_11(x, y) to be Product_{i=1..5} p_i^min{e_i, f_i}.

The sequence is the lexicographically earliest infinite sequence {a(n)} of distinct positive numbers such that, for n>2, gcd_11(a(n), a(n-1)) > 1 and gcd_11(a(n), a(n-2)) = 1.

An analog of A336957, but using only the first five primes.

Frank Stevenson has proved that a(n) always exists, something that is not true if only the primes 2, 3, 5, 7 are used. He remarks that because the small primes 13, 17, 19, ... cannot be used in the construction, some numbers take a long time to appear - are very late, in the terminology of A338053.

As can be seen from the graph, this is a much more irregular sequence than A336957.

Links

• Frank Stevenson, Table of n, a(n) for n = 1..843

Crossrefs

Cf. A336957, A338053, A338054.