cp's OEIS Frontend

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

%I A364919 #13 Sep 03 2023 10:43:59
%S A364919 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,
%T A364919 63,32,35,40,45,30,13,26,39,36,65,50,75,48,91,52,81,42,125,56,105,54,
%U A364919 121,44,99,64,143,80,117,60,77,88,147,66,169,70,135,72,17,34
%N A364919 a(0) = 1; a(n) is the smallest number m not already in the sequence such that rad(m) divides A019565(n).
%C A364919 Let k be a squarefree number and define R_k to be the set of numbers m such that rad(m) | k.
%C A364919 For n > 0, a(n) is the smallest m in R_k such that a(j) != m, j < n.
%C A364919 Conjecture: permutation of natural numbers.
%H A364919 Michael De Vlieger, <a href="/A364919/b364919.txt">Table of n, a(n) for n = 0..16384</a>
%H A364919 Michael De Vlieger, <a href="/A364919/a364919.png">Log log scatterplot of a(n)</a>, 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.
%H A364919 Michael De Vlieger, <a href="/A364919/a364919_1.png">Plot p(k)^e(k) at (x,y) = (n,k)</a>, 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.
%H A364919 Michael De Vlieger, <a href="/A364919/a364919_2.png">Fan style binary tree showing a(n)</a>, n = 0..2^12-1, with a color code similar to the scatterplot above.
%F A364919 a(2^k) = prime(k+1).
%e A364919 Let b(n) = A019565(n).
%e A364919 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.
%e A364919 a(3) = 4 since b(3) = 6, and the smallest number m such that rad(m) | 6 that has not already appeared is 4.
%e A364919 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.
%e A364919 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.
%t A364919 nn = 120; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A364919 f[x_] := Times @@ Prime@ Position[Reverse@ IntegerDigits[x, 2], 1][[All, 1]];
%t A364919 c[_] := False; c[1] = True; q[_] := 1; a[0] = 1; r[_] := 1;
%t A364919 Do[If[PrimeQ[#],
%t A364919   While[c[Set[k, #^q[#]]], q[#]++],
%t A364919   While[Or[c[r[#]], ! Divisible[#, rad[r[#]]]], r[#]++]; k = r[#] ] &[f[i]]; Set[{a[i], c[k]}, {k, True}], {i, nn}];
%t A364919 Array[a, nn + 1, 0]
%Y A364919 Cf. A005117, A007947, A019565, A289280.
%K A364919 nonn
%O A364919 0,2
%A A364919 _Michael De Vlieger_, Aug 30 2023