Square-free integers in the sequence $n^{\operatorname{rad}(n)}+\operatorname{rad}(n)+1$

Question

For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an integer $n>1$ with the definition $\operatorname{rad}(1)=1$ (the Wikipedia's article dedicated to this multiplicative function is Radical of an integer).

Question. I wondered about if there are many much or few integers $n\geq 1$ for which $$n^{\operatorname{rad}(n)}+\operatorname{rad}(n)+1\tag{1}$$ has no repeated prime factors. What work can be done, calculations, heuristics or reasonings, about if the sequence $(1)$ does contain infinitely many terms without repeated prime factors (that is integers similar than next Examples)? Many thanks.

Examples.

1) For $n=1$ the integer expressed as $(1)$ has no repeated prime factors since $1^{\operatorname{rad}(1)}+\operatorname{rad}(1)+1=3$ that is a prime number.

2) For $n=4$ one has also that the corresponding integer of the form $(1)$ is a square-free integer, that is has no repeated prime factors since $4^{\operatorname{rad}(4)}+\operatorname{rad}(4)+1=4^2+2+1=19$ that also is square-free because is a prime number.

3) For $n=19$ also $19^{\operatorname{rad}(19)}+\operatorname{rad}(19)+1=19^{19}+19+1$ is a square-free integer since has prime factorization $3\cdot 139225573\cdot 4736724757839121$.

Computational evidence. Upto $N=50$ the only integers $m's$ such that $1\leq m\leq 50$ for which $m^{\operatorname{rad}(m)}+\operatorname{rad}(m)+1$ has some repeated prime factors are $m=13,20,22$ and $m=31$.

Answer 1

Too long for a comment : Checking the primes upto $10^6$, for the following $166$ n's , we do NOT get a squarefree number :

? print(y)
[13, 20, 22, 31, 58, 67, 77, 85, 92, 94, 103, 130, 136, 139, 146, 152, 157, 160,
 164, 166, 176, 177, 185, 191, 193, 199, 200, 201, 202, 211, 212, 216, 229, 234,
 236, 238, 247, 248, 250, 260, 263, 265, 274, 277, 280, 283, 301, 304, 307, 308,
 310, 319, 325, 337, 343, 346, 350, 354, 355, 373, 377, 380, 381, 382, 391, 401,
 402, 409, 410, 414, 418, 422, 424, 427, 436, 445, 452, 454, 463, 464, 474, 481,
 484, 493, 499, 516, 517, 521, 524, 526, 535, 553, 557, 562, 567, 568, 569, 571,
 577, 581, 589, 596, 598, 607, 634, 636, 637, 643, 653, 661, 668, 670, 677, 679,
 697, 698, 706, 710, 712, 715, 733, 736, 738, 740, 742, 749, 751, 752, 761, 767,
 769, 777, 778, 787, 790, 797, 805, 812, 814, 823, 836, 851, 855, 856, 859, 863,
 877, 884, 886, 893, 895, 905, 913, 922, 931, 949, 956, 958, 967, 968, 977, 980,
 982, 985, 994, 998]
? length(y)
%48 = 166
?