OFFSET
1,2
COMMENTS
In other words, if R = A007947(a(n-2)*a(n-1)), and S (= 1 or 2) is the smallest number of terms in U divisible by any p|R, then a(n) is the least novel multiple of the greatest p|R which divides S terms in U.
When for some m > 1, a(n-1) = m*p, is the first occurrence of a multiple of prime p, then a(n) = p. The sequence after a(1) = 1 can be represented as an irregular table in which the n-th row T(n,k) starts with p = prime(n), and ends with the first subsequent term divisible by q = prime(n+1). The next row begins with q, and so on. The range of k is not always known; see Formula and Example.
Some primes (2,7,11,13,47,79,113,127,...) exhibit fully developed initial trajectories T(n,k) dominated by multiples of p, until p*q is reached, whereas for others the trajectory ends with a term m*q where m < p.
The scatterplot resembles a fine-toothed comb (as in A361133, A361534), in which each "tooth" represents the initial trajectory of a prime. It is not known whether infinitely many primes exhibit fully developed initial trajectories.
Conjectured to be a permutation of the positive integers with primes in order.
LINKS
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 prime power nor squarefree in blue, accentuating those last-mentioned whose prime power factors have multiplicities exceeding 1 in light blue.
FORMULA
The number of terms T(n,k) is <= 3*(prime(k+1) - 1)/2, with equality when the trajectory is fully developed. The number of terms in shorter trajectories is not readily known. Therefore the irregular table (see Example) is expressed as T(n), rather than T(n,k).
EXAMPLE
a(1,2) = 1,2 and 2 is the smallest p|R = 2 which divides the least number (1) of terms in U = {1,2}. Therefore a(3) = 4, is the least multiple of 2 which has not already occurred.
a(2,3) = 2,4 and since rad(2) = rad(4) = 2, S = 2 then a(4) = 6, the least multiple of 2 which has not occurred earlier.
a(3,4) = 4,6 so S = 1 since 3|6 but not 2, so a(5) = 3, since 3 has not appeared previously.
The sequence after a(1) can be shown as an irregular table of finite subsequences where each row T(n) starts with prime(n) and ends with the first subsequent multiple of prime(n+1). The table includes the fully developed trajectories of 2,7,11,13 with 3,15,24 terms respectively, each having final term in A006094. T(4) starts with 7, ends with 77 and has 3*(prime(5)-1)/2 = 15 terms.
Table T(n) for n = 1..10:
2,4,6
3,8,9,12,10
5,14
7,16,21,28,15,35,42,20,49,56,18,63,70,25,77
11,84,22,33,24,44,55,30,66,88,27,99,110,40,121,132,36,143
13,154,26,39,45,52,65,50,78,91,98,104,117,48,130,156,60,169,182,105,195,208,75,221
17,234,34,51,54,68,85,80,102,119,112,136,153,57
19,69,
23,72,46,92,138,81,115,161,126,184,207,87
29,90,58,116,174,93
MATHEMATICA
nn = 120; c[_] = False; q[_] = 1;
a[1] = i = 1; a[2] = j = 2; c[1] = c[2] = True; w = {2};
Do[If[Equal @@ w, p = Times @@ w[[1]],
g = Reverse@ Union@ Flatten[w];
p = First[
Join @@ Map[ReverseSort,
SplitBy[SortBy[
Table[{p, Count[{i, j}, _?(Divisible[#, p] &)]}, {p, g}],
Last], Last]]][[1]] ];
k = q[p]; While[c[k p], k++]; k *= p; While[c[p q[p]], q[p]++];
Set[{a[n], c[k], w, i, j},
{k, True, Append[w[[2 ;; -1]], FactorInteger[k][[All, 1]]], j, k}];
If[k == u, While[c[u], u++]], {n, 3, nn}], n];
Array[a, nn] (* Michael De Vlieger, Mar 18 2023 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David James Sycamore, Mar 18 2023
STATUS
approved