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%I #5 Nov 27 2024 18:34:35
%S 1,4,8,9,18,27,32,36,49,50,54,64,98,100,108,121,125,162,216,225,242,
%T 243,288,289,324,338,343,392,400,432,441,450,486,500,512,648,675,676,
%U 729,784,800,841,864,882,900,1000,1058,1089,1125,1152,1250,1296,1323,1350
%N Numbers m such that k = 4*m is powerful while both 4*m-1 and 4*m+1 are squarefree.
%H Michael De Vlieger, <a href="/A378172/b378172.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 1, k = 4, since both 3 and 5 are prime and thus squarefree.
%e 2 is not in the sequence since 4*2+1 = 9 is not squarefree.
%e a(2) = 4, k = 16 since both 15 and 17 are squarefree.
%e a(3) = 8, k = 32, since both 31 and 33 are squarefree.
%e a(4) = 9, k = 36, since both 35 and 37 are squarefree.
%e 16 is not in the sequence since 16*4-1 = 63 = 3^2*7 is not squarefree, etc.
%t With[{nn = 6000}, 1/4 * Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], AllTrue[# + {-1, 1}, SquareFreeQ] &] ]
%Y Cf. A001694, A005117, A335851.
%K nonn,easy
%O 1,2
%A _Michael De Vlieger_, Nov 24 2024