A290633 Lexicographically earliest sequence of positive integers such that, for any m and n > 0, gcd(a(n), a(n+1)) > 1 and a(n) != a(n+2), and if m < n then a(m) != a(n) or a(m+1) != a(n+1).

2, 2, 4, 4, 2, 6, 3, 3, 6, 2, 8, 4, 6, 6, 4, 8, 2, 10, 4, 12, 2, 14, 4, 10, 2, 12, 3, 9, 6, 8, 8, 6, 9, 3, 12, 4, 14, 2, 16, 4, 18, 2, 20, 4, 16, 2, 18, 3, 15, 5, 5, 10, 6, 12, 8, 10, 5, 15, 3, 18, 4, 20, 2, 22, 4, 24, 2, 26, 4, 22, 2, 24, 3, 21, 6, 10, 8, 12

Offset: 1

Author

Rémy Sigrist, Aug 08 2017

Keywords

• nonn
• look

Comments

a(n) > 1 for any n > 0.

If we drop the constraint "a(n) != a(n+2)", then we obtain the positive even numbers interspersed with 2's: 2, 2, 4, 2, 6, ...

Conjecturally, (a(n), a(n+1)) runs over all pairs of noncoprime positive integers; in this sense, this sequence is opposite to sequences like Stern's diatomic series (A002487).

This sequence has connections with A067992: here we avoid duplicate ordered pairs of consecutive terms, there unordered pairs, here we deal with noncoprime consecutive terms, there we (conjecturally) have coprime consecutive terms; also, the scatterplots of these sequences have similarities.

For any prime p, the sequence contains a multiple of p: by contradiction:

- let p be the least prime whose multiples are missing from the sequence (note that p > 2),

- there is only a finite number of pairs of noncoprime (p-1)-smooth numbers < p^2,

- so eventually we must have a term, say a(m), > p^2,

- if q is the least prime factor of a(m-1), then p*q would have been a better choice for a(m), hence the contradiction.

Also, if p is an odd prime, then the first multiple of p appearing in the sequence is a semiprime p*q with q < p.

If p < q are prime, then the first multiple of p appears before the first multiple of q.

For any prime p, the first occurrence of p in the sequence is immediately followed by a second occurrence of p.

For any prime p > 3:

- there is a semiprime p*q with q < p in the sequence,

- if q = 2, then this first p*q is followed by a 4,

- if q > 2, then this first p*q is followed by a 2,

- so there are infinitely many 2's or 4's in the sequence,

- if there are infinitely many 2's in the sequence, then the n-th occurrence of 2 is followed by 2*(n+e) with |e| <= 1, and every even

number appears in the sequence,

- the same conclusion applies if there are infinitely many 4's,

- hence every even number appear in the sequence.

For any n > 1, the first occurrence of n in the sequence must be either preceded or followed by the least prime factor of n (A020639).

Examples

			a(1) = 2 is suitable.
a(2) = 2 is suitable.
a(3) cannot be either 2 (=a(1)) or 3 (gcd(2,3)=1).
a(3) = 4 is suitable.
a(4) cannot be either 2 (=a(2)) or 3 (gcd(4,3)=1).
a(4) = 4 is suitable.
a(5) = 2 is suitable.
a(6) cannot be 2 (pair (2,2) already seen), 3 (gcd(2,3)=1), 4 (pair (2,4) already seen) or 5 (gcd(2,5)=1).
a(6) = 6 is suitable.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A290633
• Rémy Sigrist, Scatterplot of the first 10000 pairs of consecutive terms
• Rémy Sigrist, Colorized scatterplot of the first 100000 pairs of consecutive terms

Crossrefs

Cf. A002487, A020639, A067992.

Programs

PARI

See Links section.