OFFSET
0,2
COMMENTS
Let k be a squarefree number and define R_k to be the set of numbers m such that rad(m) | k.
For n > 0, a(n) is the smallest m in R_k such that a(j) != m, j < n.
Conjecture: permutation of natural numbers.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n), n = 0..2^14, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. We accentuate numbers in A001694 that are not prime powers with large light blue points.
Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k), n = 0..2^11, with a color function representing e(k) = 1 in black, e(k) = 2 in red, e(k) = 3 in orange, etc., and the highest e(k) in magenta. The bar at bottom indicates a(n) in a color code similar to the scatterplot above.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2^12-1, with a color code similar to the scatterplot above.
FORMULA
a(2^k) = prime(k+1).
EXAMPLE
Let b(n) = A019565(n).
a(1) = 2 since b(1) = 2. Since 2 is prime, we find the first number in the prime power range of 2 that is not in the sequence and that is 2.
a(3) = 4 since b(3) = 6, and the smallest number m such that rad(m) | 6 that has not already appeared is 4.
a(5) = 8 since b(5) = 10. R_10 begins {1, 2, 4, 5, 8, 10, 16, ...} and the smallest number m in that list that is not already in the sequence is 8.
a(6) = 9 since b(6) = 15. R_15 begins {1, 3, 5, 9, 15, 25, ...} and the smallest m in that list not already in the sequence is 9, etc.
MATHEMATICA
nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]];
c[_] := False; c[1] = True; q[_] := 1; a[0] = 1; r[_] := 1;
Do[If[PrimeQ[#],
While[c[Set[k, #^q[#]]], q[#]++],
While[Or[c[r[#]], ! Divisible[#, rad[r[#]]]], r[#]++]; k = r[#] ] &[f[i]]; Set[{a[i], c[k]}, {k, True}], {i, nn}];
Array[a, nn + 1, 0]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Aug 30 2023
STATUS
approved