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

1,1

Comments

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)

Links

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

Formula

A093616(a(n)) < n.

Mathematica

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 *)

Prog

(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

Crossrefs

Complement of A093618.

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

Sequence in context: A097319 A258211 A354929 * A089219 A102835 A095990

Adjacent sequences: A093614 A093615 A093616 * A093618 A093619 A093620

Keyword

nonn

Author

Reinhard Zumkeller, Apr 06 2004

Status

approved