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A093617
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Numbers m such that there exists a number k less than m with k*m and m^2 having an equal number of divisors.
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3
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18, 50, 75, 90, 98, 108, 126, 144, 147, 150, 198, 234, 242, 245, 294, 300, 306, 324, 338, 342, 350, 363, 384, 400, 414, 450, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 600, 605, 630, 640, 648, 650, 666, 720, 722, 726, 735, 738, 756, 774, 784, 825
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OFFSET
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1,1
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COMMENTS
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All the terms are nonsquarefree numbers (A013929).
The number k is of the form j^2*A007913(m).
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)
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LINKS
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FORMULA
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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