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In the theory of finite abstract group, abelianness-forcing number $n$ is characterized as a positive integer with standard factorization $n=p_1^{k_1}p_2^{k_2}\cdots p_r^{k_r}$ with $k_i \le 2$ and $p_i$ does not divide $p_j^{k_j}-1$ for any $1 \le i,j \le r$. I want to define a function "AbeliannessForcingNumberQ" which returns "True" if and only if the argument is a abeliannes-forcing number. But I cannot figure out how to deal with prime factors and exponents in a given number. Please help to define this function.

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abeliannessForcingNumberQ[n_Integer] := And @@ Flatten[{
   Table[Not@Divisible[Power @@ p - 1, q], {p, FactorInteger[n]}, {q, First /@ FactorInteger[n]}], 
   Thread[Last /@ FactorInteger[n] <= 2]
   }]

Select[Range[20], abeliannessForcingNumberQ]
(* 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19 *)

See also A051532.

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    $\begingroup$ What a quick and exact response! $\endgroup$
    – seoneo
    Commented Jul 9, 2019 at 23:12
  • $\begingroup$ As a minor correction, we should include $1$ as an abelianness forcing number. $\endgroup$
    – seoneo
    Commented Oct 28, 2019 at 4:24

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