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A372972
Numbers k such that A372720(k) is negative.
3
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
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]]
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Jun 02 2024
STATUS
approved