OFFSET
1,2
COMMENTS
The second condition of the definition applies iff rad(a(n-1)) = rad(a(n-2)). This occurs when n = 3 (a(2) = 2, a(3) = 4), and seems never to occur again.
Conjecture: This is a permutation of the positive integers, with primes in order. Each prime p generates a trajectory T(p), dominated by multiples of p, see graph. Typically T(prime(k)) is initiated following the earliest term divisible by prime(k). Some trajectories are more prominent in the graph than others e.g. T(7),T(11),T(13) extend from p = prime(k) to prime(k+1)*prime(k). Others, e.g. T(3) terminate earlier but include prime(k)^2, whilst some do not reach as far as prime(k)^2 (T(5) has just two terms). Thus three categories of prime trajectory can be distinguished in the graph (respectively "full", "medium" and "small"). The graph resembles an inclined comb whose teeth correspond to full and medium trajectories; see Example.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue.
FORMULA
If T(prime(k)) is a fully developed trajectory is contains (prime(k+1) - 1)*3/2 terms.
EXAMPLE
a(1,2) = 1,2 so a(3) = 4, the least novel multiple of 2 (which divides 2 but not 1). Since rad(2) = rad(4) = 2 there is no prime which divides one of a(2), a(3) but not the other so by the second condition of the definition a(4) = 3, the least novel multiple of the smallest prime (3) which divides neither a(2) = 2 nor a(3) = 4.
The sequence can be presented as an irregular table where row(k) starts with A008578(k), and with the exception of rows 1 and 2, ends with the earliest multiple of A008578(k+1).
The table starts:
1;
2,4;
3,6,8,9,12,10;
5,14;
7,16,21,28,15,35,42,20,49,56,18,63,70,25,77;
11,84,22,33,24,44,55,30,66,88,27,110,40,121,132,36,143;
13,154.....
T(3) is a medium trajectory, includes 3^2 but not 3*5 = 15, which appears later, in T(7). T(5) is a short trajectory, stopped by 14, does not include 25 which is delayed until T(7); T(7) is the first full trajectory, including 49, and ending with 7*11 = 77. In full and medium trajectories T(p) we see pairs of consecutive multiples of p separated by a multiple of a smaller prime. If T(prime(k)) is full it contains (prime(k+1) - 1) multiples of prime(k) and (prime(k+1) - 1)/2 multiples of smaller primes, thus T(7) contains 15 terms; see Formula.
MATHEMATICA
nn = 120;
c[_] := False; m[_] := 1;
f[x_] := f[x] = FactorInteger[x][[All, 1]];
g[x_] :=
Block[{q = 2},
If[OddQ[x], q, While[Divisible[x, q], q = NextPrime[q] ]; q] ];
Array[Set[{a[#], c[#]}, {#, True}] &, 2];
i = a[1]; j = a[2]; u = 3; ri = {}; rj = {j};
Do[Set[r, Times @@ Union[ri, rj]]; If[ri == rj,
While[c[Set[k, # m[#] ] ], m[#]++] &[g[r]],
While[c[Set[k, # m[#] ] ], m[#]++] &[FactorInteger[r][[-1, 1]] ] ];
Set[{a[n], c[k], i, j, ri, rj}, {k, True, j, k, rj, f[k]}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Nov 06 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore and Michael De Vlieger, Nov 06 2023
STATUS
approved