A378172 Numbers m such that k = 4*m is powerful while both 4*m-1 and 4*m+1 are squarefree.

1, 4, 8, 9, 18, 27, 32, 36, 49, 50, 54, 64, 98, 100, 108, 121, 125, 162, 216, 225, 242, 243, 288, 289, 324, 338, 343, 392, 400, 432, 441, 450, 486, 500, 512, 648, 675, 676, 729, 784, 800, 841, 864, 882, 900, 1000, 1058, 1089, 1125, 1152, 1250, 1296, 1323, 1350

Offset

1,2

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

a(1) = 1, k = 4, since both 3 and 5 are prime and thus squarefree.
2 is not in the sequence since 4*2+1 = 9 is not squarefree.
a(2) = 4, k = 16 since both 15 and 17 are squarefree.
a(3) = 8, k = 32, since both 31 and 33 are squarefree.
a(4) = 9, k = 36, since both 35 and 37 are squarefree.
16 is not in the sequence since 16*4-1 = 63 = 3^2*7 is not squarefree, etc.

Mathematica

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] &] ]

Crossrefs

Cf. A001694, A005117, A335851.

Sequence in context: A116030 A116020 A354869 * A213015 A064393 A372034

Adjacent sequences: A378169 A378170 A378171 * A378173 A378174 A378175

Keyword

nonn,easy

Author

Michael De Vlieger, Nov 24 2024

Status

approved