Is the following conjecture obviously false? I have checked on a computer for $n$ up to $200$. Since that is pretty low, and there doesn't seem to be any good upper bound on the minimum value of $t$ needed, I wouldn't be surprised if it was still false.
Let $n > 1$ be squarefree. Then there exists a $t \geq n$ such that all the numbers $$n + t^2 - k^2, \qquad 0 \leq k < t$$ have squarefree part greater than $n$.
Of course I would love to kill this with something like Schinzel's hypothesis H, but that doesn't seem to be enough since the list of polynomials involved grows with the input.
Here is the list of the minimal $t \geq n$ that works for each squarefree $1 < n \leq 203$, organized as ordered triples $(n, t, s)$, where $t$ is the minimum required $t$, and $s$ is the smallest squarefree part of all the $n + t^2 - k^2$. At first I was hopeful that one could choose $t$ close to $n$, but some of the data points actually need very large $t$. Perhaps the thing to do is to parameterize $t = an + b$ with $b < n$, and hopefully observe that $a = 1$ and small $b$ works a lot of the time.
(2, 2, 5)
(3, 6, 14)
(5, 5, 14)
(6, 6, 17)
(7, 8, 22)
(10, 10, 29)
(11, 12, 34)
(13, 13, 38)
(14, 14, 41)
(15, 30, 74)
(17, 30, 19)
(19, 22, 62)
(21, 21, 62)
(22, 25, 71)
(23, 30, 82)
(26, 27, 79)
(29, 33, 94)
(30, 30, 89)
(31, 32, 94)
(33, 78, 41)
(34, 34, 101)
(35, 70, 174)
(37, 85, 206)
(38, 38, 113)
(39, 108, 62)
(41, 41, 122)
(42, 126, 69)
(43, 46, 55)
(46, 46, 137)
(47, 48, 67)
(51, 54, 158)
(53, 57, 166)
(55, 110, 274)
(57, 95, 86)
(58, 62, 181)
(59, 64, 71)
(61, 93, 66)
(62, 67, 78)
(65, 65, 194)
(66, 77, 74)
(67, 74, 214)
(69, 93, 254)
(70, 91, 103)
(71, 122, 314)
(73, 143, 358)
(74, 84, 86)
(77, 135, 106)
(78, 390, 114)
(79, 92, 262)
(82, 95, 271)
(83, 92, 266)
(85, 187, 149)
(86, 86, 257)
(87, 135, 89)
(89, 137, 122)
(91, 148, 130)
(93, 177, 138)
(94, 110, 178)
(95, 120, 334)
(97, 355, 193)
(101, 101, 302)
(102, 2346, 134)
(103, 112, 326)
(105, 2947, 145)
(106, 154, 157)
(107, 176, 458)
(109, 113, 334)
(110, 143, 167)
(111, 252, 219)
(113, 145, 402)
(114, 300, 137)
(115, 320, 146)
(118, 128, 373)
(119, 206, 146)
(122, 150, 421)
(123, 222, 131)
(127, 166, 458)
(129, 207, 146)
(130, 845, 139)
(131, 208, 194)
(133, 427, 206)
(134, 158, 449)
(137, 161, 218)
(138, 462, 237)
(139, 148, 434)
(141, 159, 173)
(142, 191, 163)
(143, 268, 182)
(145, 377, 226)
(146, 162, 181)
(149, 307, 158)
(151, 244, 166)
(154, 322, 173)
(155, 160, 179)
(157, 205, 178)
(158, 225, 166)
(159, 742, 183)
(161, 189, 538)
(163, 338, 838)
(165, 3555, 506)
(166, 188, 541)
(167, 481, 239)
(170, 961, 311)
(173, 235, 642)
(174, 212, 206)
(177, 954, 213)
(178, 202, 197)
(179, 277, 183)
(181, 359, 229)
(182, 226, 233)
(183, 732, 219)
(185, 435, 274)
(186, 186, 557)
(187, 850, 238)
(190, 535, 359)
(191, 216, 211)
(193, 727, 241)
(194, 244, 206)
(195, 880, 211)
(197, 355, 218)
(199, 386, 259)
(201, 471, 218)
(202, 229, 659)
(203, 212, 218)