A372972 Numbers k such that A372720(k) is negative.

162, 250, 324, 384, 486, 648, 686, 768, 972, 1152, 1250, 1296, 1372, 1458, 1536, 1728, 1875, 1944, 2058, 2250, 2304, 2430, 2500, 2560, 2592, 2662, 2738, 2916, 3000, 3072, 3362, 3402, 3456, 3698, 3750, 3840, 3888, 3993, 4050, 4116, 4374, 4394, 4418, 4500, 4608

Offset

1,1

Comments

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.

For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).

For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.

Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.

In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.

For s > 1, an infinite number of k in R are such that g(k) is negative. For example, with s = 6, all terms k > 864 in A033845 are in this sequence.

Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

a(1) = 162 = 2*3^4, since tau(162) - f(162)
     = (1+1)*(4+1) - card(A369609(162))
     = 10 - 12 = -2.
a(2) = 250 = 2*5^3, since tau(250) - f(250)
     = (1+1)*(3+1) - card(A369609(250))
     = 8 - 9 = -1.
a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
     = (2+1)*(4+1) - card(A369609(324))
     = 15 - 16 = -1, etc.

Mathematica

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], _?(Divisible[r, rad[#]] &)], {n, 5000}], _?(# < 0 &)][[All, 1]]

Crossrefs

Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.

Sequence in context: A259220 A107120 A045010 * A085446 A274027 A206210

Adjacent sequences: A372969 A372970 A372971 * A372973 A372974 A372975

Keyword

nonn

Author

Michael De Vlieger, Jun 02 2024

Status

approved