A364919 a(0) = 1; a(n) is the smallest number m not already in the sequence such that rad(m) divides A019565(n).

1, 2, 3, 4, 5, 8, 9, 6, 7, 14, 21, 12, 25, 10, 15, 16, 11, 22, 27, 18, 55, 20, 33, 24, 49, 28, 63, 32, 35, 40, 45, 30, 13, 26, 39, 36, 65, 50, 75, 48, 91, 52, 81, 42, 125, 56, 105, 54, 121, 44, 99, 64, 143, 80, 117, 60, 77, 88, 147, 66, 169, 70, 135, 72, 17, 34

Offset: 0

Author

Michael De Vlieger, Aug 30 2023

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A005117, A007947, A019565, A289280.

Programs

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]

Formula

a(2^k) = prime(k+1).