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%I #23 Apr 05 2024 20:09:35
%S 12,20,28,44,52,60,68,76,84,92,116,124,132,140,148,156,164,172,188,
%T 204,212,220,228,236,244,260,268,276,284,292,308,316,332,340,348,356,
%U 364,372,380,388,404,412,420,428,436,444,452,460,476,492,508,516,524,532,548
%N Numbers k neither squarefree nor prime power such that both rad(k)*A119288(k) > k and rad(k)*A053669(k) > k.
%C Subset of A126706, numbers that are neither squarefree nor prime powers.
%C For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
%C A355432(k) = A360543(k) = 0. There exist neither nondivisor m < k such that rad(m) = rad(k), nor m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k.
%C Apparently this is A081770 without the leading 4. - _R. J. Mathar_, Sep 05 2023
%C From _Peter Munn_, Mar 05 2024: (Start)
%C The preceding observation is true for the whole sequence, for reasons outlined below.
%C To qualify for this sequence, a number k must be smaller than 2 different multiples of rad(k): one based on a divisor, A119288(k): the other on a nondivisor, A053669(k).
%C For k that is not a prime power, straightforward calculations show (1) if k = 2 * rad(k) then k satisfies both of these comparisons, whereas (2) for k >= 3 * rad(k), k fails the divisor-based comparison if k is a multiple of 6 and fails the nondivisor-based comparison otherwise.
%C (End)
%H Michael De Vlieger, <a href="/A364999/b364999.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael De Vlieger, <a href="/A364999/a364999.png">Annotated plot of b(n) = A126706(n)</a>, with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364998 in gold, in A364997 in green, and in A361098 in red.
%H Michael De Vlieger, <a href="/A364999/a364999_1.png">Plot of b(n)</a>, with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms. This uses the same color scheme as described immediately above.
%H Michael De Vlieger, <a href="/A364999/a364999_2.png">Plot of b(n)</a>, with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates fairly constant density of a(n) in A126706 as well as a slight quasiperiodic pattern approximately mod 169.
%F Intersection of A363082 and A360767.
%F From _Peter Munn_, Feb 21 2024: (Start)
%F a(n) = 2*A039956(n+1).
%F Asymptotic density is 1/Pi^2 = 0.101321183642337... (A092742). (End)
%F From _Michael De Vlieger_, Mar 08 2024: (Start)
%F {a(n)} = A366825 \ A366460, i.e., even terms in A366825.
%F A088860 = {a(n)} intersect A025487 = {a(n)} intersect A055932, where A088860(k) = 2*A002110(k). (End)
%e Let b(n) = A126706(n), S = A360767, and T = A363082.
%e b(1) = a(1) = 12 since p*r = 3*6 = 18 and q*r = 5*6 = 30, and both exceed 12. Indeed, 12 is in both S and T.
%e b(2) = 18 is not in the sequence since p*r = 3*6 = 18; 18 is not in S.
%e b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36.
%e b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, and is not in T, etc.
%t Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]
%Y Cf. A007947, A039956, A053669, A081770, A088860, A092742, A119288, A126706, A355432, A360432, A360767, A361098, A363082, A364998, A364999.
%K nonn
%O 1,1
%A _Michael De Vlieger_, Aug 16 2023