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

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 $m=31$.

• Everyone I am asking about feedback about this sequence (I suspect that it is very difficult to solve), thus I am going to accept an answer in the spirit as was asked my Question (what work can be done about the study if there are infinitely many square-free integers of the form $n^{\operatorname{rad}(n)}+\operatorname{rad}(n)+1$) telling us a fruitful approach or a remarkable heuristic about our problem. Many thanks. – user243301 Jun 4 '18 at 14:22
• Just an observation: this set contains all numbers of the type $$p^{kp}+p+1,$$ with $p$ prime and $k$ positive integer. In particular, with $p=2$, it contains all numbers of the form $x_k:=2^k+3$. It has been shown in sciencedirect.com/science/article/pii/S0022314X17303943 that $\omega(x_n)$ is not bounded. However, the quantitative estimate deriving from that result is not sufficient to conclude anything about the multiplicity of its prime factors. – Paolo Leonetti Jun 4 '18 at 14:36
• Many thanks for share your reasoning @PaoloLeonetti I am asking it as an amateur. Feel free to add an answer if in next future you want to add your remarks/observations as a contribution. – user243301 Jun 4 '18 at 15:14
• It seems that about one out of six $n's$ gives a number which is NOT squarefree. I checked for the first $10^5$ n's whether the given expression is divisible by the square of a prime less than $10^4$ and for $16\ 671$ n's it is. – Peter Jun 7 '18 at 14:22
• Of the first $10^4$ n's , checking the prime upto $10^5$, I found $1\ 665$ examples not being squarefree – Peter Jun 7 '18 at 15:25

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
?

• Many thanks for the data. I hope that with your help or the help or other users we can do solve the Question. – user243301 Jun 7 '18 at 16:28