A093616 -id:A093616 - cp's OEIS frontend

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.

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

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

From Amiram Eldar, Apr 15 2024: (Start)
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)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..300 from Vincenzo Librandi)

Complement of A093618.

Cf. A000005, A000290, A007913, A013929, A048691, A093616.

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 *)

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

A093616(a(n)) < n.

A093618 Numbers m such that for all k less than m the number of divisors of k*m and m^2 are different.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1800 from Vincenzo Librandi)

Complement of A093617.

Cf. A000005, A000290, A048691, A093616.

A093616[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Select[Range[100], A093616[#] == # &] (* Jean-François Alcover, Aug 14 2014 *)

is(n) = {my(k = 1, d = numdiv(n^2)); while(k < n && numdiv(k*n) != d, k++); k == n;} \\ Amiram Eldar, Apr 15 2024

A093616(a(n)) = n.

Corrected by Franklin T. Adams-Watters, Dec 19 2006

cp's OEIS Frontend

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.

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