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A069056 Numbers k such that Sum_{d|k} d^2*mu(d) divides k^2.

%I A069056 #29 Apr 20 2025 02:33:23
%S A069056 1,12,24,36,48,72,96,108,120,144,192,216,240,288,324,336,360,384,432,
%T A069056 480,576,600,648,672,720,768,864,960,972,1008,1080,1152,1200,1296,
%U A069056 1344,1440,1536,1728,1800,1920,1944,2016,2160,2304,2352,2400,2448,2592,2688
%N A069056 Numbers k such that Sum_{d|k} d^2*mu(d) divides k^2.
%C A069056 Numbers k such that A046970(k) divides k^2. [corrected by _Amiram Eldar_, Apr 20 2025]
%C A069056 If n > 1, a(n+1) - a(n) == 0 (mod 12), so a(n+1) - a(n) = 12 for n=2,3,4,5,7,8,...; a(n+1) - a(n) = 24 for n=6,9,.... Conjecture: if c > 2 and n > 1, Sum_{d|n} d^c*mu(d) never divides n^c. Hence A063453(n) never divides n^3 for n > 1.
%H A069056 Amiram Eldar, <a href="/A069056/b069056.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Reinhard Zumkeller)
%t A069056 f[d_] := d^2*MoebiusMu[d]; ok[n_] := Divisible[n^2, Total[f /@ Divisors[n]]]; Select[Range[3000], ok] (* _Jean-François Alcover_, Nov 15 2011 *)
%t A069056 q[k_] := Divisible[k^2, Times @@ ((First[#]^2-1)& /@ FactorInteger[k])]; q[1] = True; Select[Range[3000], q] (* _Amiram Eldar_, Apr 20 2025 *)
%o A069056 (PARI) for(n=1,1000,if(n^2%sumdiv(n,d,moebius(d)*d^2)==0,print1(n,",")))
%o A069056 (PARI) isok(k) = {my(f = factor(k)); !(k^2 % prod(i = 1, #f~, f[i,1]^2-1));} \\ _Amiram Eldar_, Apr 20 2025
%o A069056 (Haskell)
%o A069056 a069056 n = a069056_list !! (n-1)
%o A069056 a069056_list = filter (\x -> x ^ 2 `mod` a046970 x == 0) [1..]
%o A069056 -- _Reinhard Zumkeller_, Jan 19 2012
%Y A069056 Cf. A046970, A063453.
%K A069056 easy,nice,nonn
%O A069056 1,2
%A A069056 _Benoit Cloitre_, Apr 07 2002