A093618 -id:A093618 - cp's OEIS frontend

A093616 a(n) is the smallest k such that k*n has exactly as many divisors as n^2.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 8, 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, 8, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000290, A048691, A093617, A093618.

a[n_] := For[k = 1, True, k++, If[DivisorSigma[0, k*n] == DivisorSigma[0, n^2], Return[k]]]; Array[a, 72] (* Jean-François Alcover, Aug 14 2014 *)

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

A000005(a(n)*n) = A000005(n^2) and A000005(m*n) <> A000005(n^2) for m < a(n).

a(A093617(n)) < n, a(A093618(n)) = 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.

Original entry on oeis.org

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.

cp's OEIS Frontend

A093616 a(n) is the smallest k such that k*n has exactly as many divisors as n^2.

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