|
| |
|
|
A356850
|
|
a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number not occurring earlier such that a(n) is coprime to the previous Omega(a(n)) terms.
|
|
5
|
|
|
|
1, 2, 3, 5, 4, 7, 9, 10, 11, 13, 6, 17, 19, 14, 15, 23, 22, 21, 25, 26, 29, 27, 31, 8, 33, 35, 34, 37, 39, 38, 41, 43, 45, 28, 47, 51, 46, 49, 53, 55, 12, 59, 61, 57, 20, 67, 69, 58, 65, 71, 62, 63, 73, 74, 77, 75, 79, 52, 83, 85, 81, 44, 89, 87, 82, 91, 93, 86, 95, 97, 94, 99, 101, 103, 50, 107
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The terms are concentrated along lines that contain numbers with a lowest prime factor of 2 or 3. Two of these lines are initially separated but join after approximately 130000 terms. This combined line then joins the uppermost line, which contains numbers with all prime factors and has a gradient of ~1.59, after approximately 680000 terms at which point a new series of smaller values appears. See the linked images.
Numbers with a larger number of prime divisors relative to the numbers close to it appear much later in the sequence. For example a(4014) = 16, a(14219) = 40, while even more delayed are a(685301) = 24 and a(704634) = 36. These last two appear after the above mentioned line merging after 680000 terms.
The lower lines containing terms with primes factors of 2 and 3 visible in the image of terms up to 1000000 are curving upward, possibly repeating the earlier behavior seen where similar lines eventually join with the uppermost line. If these do in fact eventually reach the uppermost line it is plausible this will once again signal the start of a new series of much lower valued terms.
The two lowest unseen numbers after 1000000 terms are 32 and 48, for former indicating that the longest run of consecutive odd values is only four after 1000000 terms. Although the sequence is conjectured to be a permutation of the positive integers, if these missing terms, especially those of the form 2^k, only appear after the recombination of the lower lines with the upper line then it may take an extraordinarily large number of terms for some values, like 2^k for large k, to eventually appear.
In the first 1000000 terms the only fixed points are the first three terms along with 14 and 15. It is possible more exist if the above mentioned upward trend of smaller values does occur.
|
|
|
LINKS
|
Michael De Vlieger, Annotated log-log scatterplot of a(n) n = 1..2^15, showing records in red, local minima in blue, highlighting primes in green, other prime powers in gold, and numbers neither prime powers nor squarefree in light blue.
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^15, with a color function according to Omega(a(n)), where red = 1, amber = 2, ..., dark blue = 6.
Scott R. Shannon, Image for n=1..100000 with color. Terms with lowest prime factor of 2 are show in red, those with 3 in yellow, those with 5 in green, and all others in white.
|
|
|
EXAMPLE
|
a(7) = 9 as Omega(9) = A001222(9) = 2, and 9 is coprime to the previous two terms, namely a(6) = 7 and a(5) = 4.
|
|
|
MATHEMATICA
|
nn = 120; c[_] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = 3; r = 2; m = 1; Do[k = u; If[# > m, m = #] &@ Floor@ Log2[r]; Do[If[i == 1, p[1] = a[n - i], Set[p[i], p[i - 1]*a[n - i]]], {i, m}]; While[Nand[! c[k], CoprimeQ[k, p[PrimeOmega[k]]]], k++]; Set[{a[n], c[k]}, {k, True}]; If[k == u, While[c[u], u++]]; If[k > r, r = k], {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Nov 07 2022 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
