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A093617 Numbers m such that there exists a number k less than m with k*m and m^2 having an equal number of divisors.

%I A093617 #21 Apr 15 2024 03:35:11
%S A093617 18,50,75,90,98,108,126,144,147,150,198,234,242,245,294,300,306,324,
%T A093617 338,342,350,363,384,400,414,450,490,500,507,522,525,540,550,558,578,
%U A093617 588,600,605,630,640,648,650,666,720,722,726,735,738,756,774,784,825
%N A093617 Numbers m such that there exists a number k less than m with k*m and m^2 having an equal number of divisors.
%C A093617 From _Amiram Eldar_, Apr 15 2024: (Start)
%C A093617 All the terms are nonsquarefree numbers (A013929).
%C A093617 The number k is of the form j^2*A007913(m).
%C A093617 The numbers of terms that do not exceed 10^k, for k = 1, 2, ..., are 0, 5, 64, 678, 6954, 69867, 699511, 6996322, 69962916, 699616048, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06996... . (End)
%H A093617 Amiram Eldar, <a href="/A093617/b093617.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..300 from Vincenzo Librandi)
%F A093617 A093616(a(n)) < n.
%t A093617 A093616[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Select[Range[1000], A093616[#] < # &] (* _Jean-François Alcover_, Aug 14 2014 *)
%t A093617 f[p_, e_] := p^(e + Mod[e, 2]); q[n_] := Module[{fct = FactorInteger[n], d, m, k = 1}, d = Times @@ ((2*# + 1) & /@ fct[[;; , 2]]); s = Times @@ f @@@ fct; m = Sqrt[n^2/s]; While[k < m && DivisorSigma[0, k^2*s] != d, k++]; k < m]; Select[Range[1000], q] (* _Amiram Eldar_, Apr 15 2024 *)
%o A093617 (PARI) is(n) = {my(f = factor(n), d = prod(i = 1, #f~, 2*f[i, 2] + 1), s = prod(i = 1, #f~, f[i, 1]^(f[i, 2] + f[i, 2]%2)), m = sqrtint(n^2/s), k = 1); while(k < m && numdiv(k^2 * s) != d, k++); k < m;} \\ _Amiram Eldar_, Apr 15 2024
%Y A093617 Complement of A093618.
%Y A093617 Cf. A000005, A000290, A007913, A013929, A048691, A093616.
%K A093617 nonn
%O A093617 1,1
%A A093617 _Reinhard Zumkeller_, Apr 06 2004