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%I #29 Jun 03 2024 18:32:43
%S 162,250,324,384,486,648,686,768,972,1152,1250,1296,1372,1458,1536,
%T 1728,1875,1944,2058,2250,2304,2430,2500,2560,2592,2662,2738,2916,
%U 3000,3072,3362,3402,3456,3698,3750,3840,3888,3993,4050,4116,4374,4394,4418,4500,4608
%N Numbers k such that A372720(k) is negative.
%C Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
%C For squarefree k, A372720(k) >= 0, since A008479(k) = 1 while tau(k) = 2^omega(k).
%C For prime power p^m, A372720(p^m) = 1, since A008479(p^m) = m while tau(k) = m+1.
%C Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
%C 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.
%C 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.
%C 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.
%H Michael De Vlieger, <a href="/A372972/b372972.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 162 = 2*3^4, since tau(162) - f(162)
%e = (1+1)*(4+1) - card(A369609(162))
%e = 10 - 12 = -2.
%e a(2) = 250 = 2*5^3, since tau(250) - f(250)
%e = (1+1)*(3+1) - card(A369609(250))
%e = 8 - 9 = -1.
%e a(3) = 324 = 2^2*3^4, since tau(324) - f(324)
%e = (2+1)*(4+1) - card(A369609(324))
%e = 15 - 16 = -1, etc.
%t 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]]
%Y Cf. A000005, A001221, A007947, A008479, A126706, A372720, A372864.
%K nonn
%O 1,1
%A _Michael De Vlieger_, Jun 02 2024
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