The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A364280 Lexicographically earliest sequence of distinct positive integers such that a(n) is the least novel multiple of m, the product of all primes q < gpf(a(n-2)*a(n-1)) which do not divide a(n-2)*a(n-1); a(1) = 1, a(2) = 2. 1
1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 105, 12, 13, 385, 18, 14, 15, 16, 17, 15015, 20, 19, 51051, 30, 21, 22, 25, 42, 23, 230945, 84, 24, 35, 26, 33, 70, 27, 28, 40, 36, 29, 37182145, 48, 31, 1078282205, 54, 32, 34, 30030, 37, 6678671, 60060, 38, 51, 5005, 44, 39 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
It follows from the definition that the sequence is infinite.
Let r(n) = a(n-2)*a(n-1)).
If rad(r(n)) is a primorial, then every prime q < gpf(r(n)) divides r(n), so m = 1, the empty product, and a(n) = u, the smallest missing number in the sequence so far.
If rad(r(n)) is not a primorial, then m > 1, and significant spikes can occur in scatterplot when there are multiple primes < gpf(r(n)) which do not divide r(n) (e.g., a(12) = 105, a(15) = 385, a(21) = 15015).
The only way a prime can occur is as u.
The sequence is a permutation of the positive integers since no number appears more than once and m = 1 eventually introduces any number not already placed consequent to terms arising from m > 1.
LINKS
Michael De Vlieger, Log log scatterplot of log_10(a(n)), n = 1..2^16, highlighting prime a(n) in red.
EXAMPLE
a(4) = 4, a(5) = 5, and 3 is the only prime < 5 which does not divide 20, therefore m = 3 and a(6) = 6 since 3 has occurred once already.
a(10) = 8, a(11) = 11 and the product of all primes < 11 which do not divide 8*11 = 88 is 3*5*7 = 105, which has not occurred earlier, so a(12) = 105.
MATHEMATICA
nn = 120; c[_] := False; m[_] := 1; a[1] = i = 1; a[2] = j = 2; c[1] = c[2] = True;
f[x_] := Times @@ Complement[Prime@ Range[PrimePi@ #[[-1]] - 1], #] &[
FactorInteger[x][[All, 1]]];
Do[While[Set[k, f[i j]]; c[k m[k]], m[k]++]; k *= m[k];
Set[{a[n], c[k], i, j}, {k, True, j, k}], {n, 3, nn}];
Array[a, nn] (* Michael De Vlieger, Jul 17 2023 *)
CROSSREFS
Sequence in context: A266645 A372368 A266646 * A361810 A362854 A102455
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Michael De Vlieger, Jul 17 2023
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 15 03:12 EDT 2024. Contains 373402 sequences. (Running on oeis4.)