A362041 - OEIS

%I #29 May 19 2023 04:24:38

%S 1,2,4,3,6,8,12,10,18,5,9,14,16,24,20,22,15,36,7,40,26,48,28,30,21,32,

%T 34,54,45,38,50,27,42,44,60,46,72,56,33,80,52,96,98,58,39,90,11,62,25,

%U 84,64,66,75,68,70,108,63,74,120,76,51,78,100,144,82,126,13,57,86,35,88,150,92,94,147,162

%N a(0) = 1; for n > 0, a(n) is the largest k < A013929(n) such that rad(k) = rad(A013929(n)), where rad(n) = A007947(n).

%C Permutation of natural numbers.

%C Let m = A013929(n) and let R_m be the sequence of numbers k such that rad(k) = rad(m). a(n) gives the predecessor of m in R_m.

%H Michael De Vlieger, <a href="/A362041/b362041.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael De Vlieger, <a href="/A362041/a362041.png">Log log scatterplot of a(n)</a>, n = 0..2^16

%H Michael De Vlieger, <a href="/A362041/a362041_1.png">Log log scatterplot of a(n)</a>, n = 0..2^12, showing primes in red, composite prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue. Numbers k with omega(k) > 1 and all exponents exceeding 1 are highlighted in large light blue dots.

%F A013929(n) = p^e, a prime power, e > 0, implies a(n) = p^(e-1).

%F A013929(n) = p^2 implies a(n) = p.

%e A013929(1) = 4; the smallest k < 4 such that rad(k) = rad(4) = 2 is a(1) = 2.

%e A013929(2) = 8; the smallest k < 8 such that rad(k) = rad(8) = 2 is a(2) = 4.

%e A013929(3) = 9; the smallest k < 9 such that rad(k) = rad(9) = 3 is a(3) = 3.

%e A013929(4) = 12; the smallest k < 12 such that rad(k) = 6 is a(4) = 6.

%t rad[x_] := Times @@ FactorInteger[x][[All, 1]]; {1}~Join~Table[Function[r, SelectFirst[Range[m - 1, 1, -1], r == rad[#] &] ][rad[m]], {m, Select[Range[225], Not @* SquareFreeQ]}]

%Y Cf. A007947, A013929, A289280.

%K nonn

%O 0,2

%A _Michael De Vlieger_, May 01 2023