A362044 - OEIS

A362044

a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

%I #11 Apr 19 2023 19:00:42

%S 32,80,128,135,343,352,512,864,891,1088,875,1216,1053,1728,2048,2187,

%T 1375,2187,2048,2048,3125,4224,2187,4802,4736,3773,5832,5248,4913,

%U 5504,7047,4459,7533,8192,6859,10368,10935,8192,11264,8991,12312,12167,8192,5831,8192,9963,10449,16640,16807,17152,18432

%N a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

%C The largest k such that k < p^2 such that p is prime and rad(k) | p is p itself.

%H Michael De Vlieger, <a href="/A362044/b362044.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A362044/a362044.png">Scatterplot of a(n), m^2, and b(n)</a>, n = 1..2^14, where b(n) = A362045(n) is shown in red, m^2 in black, and a(n) in blue.

%e a(1) = 32 since m = 6 and the largest k < m^2 such that rad(k) | 6 is 32. This is to say, the number that precedes 6^2 in A003586 is 32.

%e a(2) = 80 since m = 10 and the largest k < m^2 such that rad(k) | 10 is 80. This is to say, the number that precedes 10^2 in A003592 is 80.

%e Table of n = 1..12, m = A120944(n), a(n), and m^2.

%e n m a(n) m^2

%e ---------------------

%e 1 6 32 36

%e 2 10 80 100

%e 3 14 128 196

%e 4 15 135 225

%e 5 21 343 441

%e 6 22 352 484

%e 7 26 512 676

%e 8 30 864 900

%e 9 33 891 1089

%e 10 34 1088 1156

%e 11 35 875 1225

%e 12 38 1216 1444

%t Table[m = k^2 - 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m--]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]

%Y Cf. A007947, A120944, A289280, A362045.

%K nonn

%O 1,1

%A _Michael De Vlieger_, Apr 05 2023