OFFSET
1,1
COMMENTS
Does there exist an a(n+1) that is coprime to a(n) for any n?
Conjecture: a(n+1) is never coprime to a(n). This holds for n = 1..2^20. This would imply that this variant does not feature blocks in the sense they are defined for A280864.
In A280864, no block (determined by partitioning the sequence according to a(n-1) coprime to a(n)) ends in a squarefree composite number, thus the reason for restraining this sequence within A120944.
There exist long runs of odd terms that eventually break, admitting many even squarefree semiprimes in an episodic manner (see scatterplot). Within the first 2^21 terms, the longest run of odd terms is 388, beginning with a(318575) = 47*5639 and ending with a(318962) = 287691 = 3*17*5641.
Even squarefree semiprimes dominate local minima, while local maxima have a relatively high number of prime factors (omega = A001221).
The smallest missing number, given 2^21 terms, is 27418 = 2*13709, which emerged at a(1533358).
Apart from 15 and 21, the smallest missing number is an even squarefree semiprime for n <= 2^21. The sequence undergoes episodes of "cleaning up", picking up many missing numbers. The most conspicuous large scale example is the interval a(707776) = 312541 = 53*5897 through a(725184) = 575633 = 53*10861, with minimum: a(707798) = 12086 = 2*6043. This particular interval also manifests a 128-term long recurrence regarding number of prime factors. Conditions of onset and termination of these sort of intervals remain unknown.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20, with a color function where black represents even squarefree semiprimes, red = odd squarefree semiprimes, orange = sphenic numbers, yellow = a(n) such that omega(a(n)) = 4, .., blue = a(n) such that omega(a(n)) = 7.
Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 5X vertical exaggeration.
Michael De Vlieger, Plot omega(a(n)) at (x,y) = (n mod 1024, floor(n/1024)), with a color function as above, including showing even squarefree semiprimes in black.
EXAMPLE
Table of n, a(n) for n = 1..16:
n a(n) Prime factors
-----------------------------------
1 6 2 3
2 30 2 3 5
3 35 5 7
4 14 2 7
5 10 2 5
6 15 3 5
7 21 3 7
8 70 2 5 7
9 110 2 5 11
10 33 3 11
11 39 3 13
12 26 2 13
13 22 2 11
14 55 5 11
15 65 5 13
16 78 2 3 13
MATHEMATICA
Block[{c, i, j, k, m, w, nn},
nn = 120; c[_] := False; c[1] = True; i = j = 1;
Reap[Do[
w = GCD[i, j]; m = j/w; k = m;
While[Or[c[k], PrimeQ[k], ! SquareFreeQ[k], ! CoprimeQ[w, k] ], k += m];
Sow[k]; Set[{c[k], i, j}, {True, j, k}],
{n, nn}] ][[-1, 1]] ]
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Michael De Vlieger, Jun 25 2026
STATUS
approved