|
%I #15 Feb 24 2024 11:07:38
%S 36,48,50,54,72,75,80,96,98,100,108,112,135,144,147,160,162,189,192,
%T 196,200,216,224,225,240,242,245,250,252,270,288,294,300,320,324,336,
%U 338,350,352,360,363,375,378,384,392,396,400,405,416,432,441,448,450,468,480,484,486,490,500,504,507,525
%N Intersection of A360765 and A360768.
%C Numbers k that are neither prime powers nor squarefree, such that rad(k) * A053669(k) < k and k/rad(k) >= A119288(k), where rad(k) = A007947(k).
%C Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive.
%C Subset of A126706. All terms are neither prime powers nor squarefree.
%C From _Michael De Vlieger_, Aug 03 2023: (Start)
%C Superset of A286708 = A001694 \ {{1} U A246547}, which in turn is a superset of A303606. We may write k in A286708 as m*rad(k)^2, m >= 1. Since omega(k) > 1, it is clear both k/rad(k) > A053669(k) and k/rad(k) >= A119288(k). Also superset of A359280 = A286708 \ A303606.
%C This sequence contains {A002182 \ A168263}. (End)
%H Michael De Vlieger, <a href="/A361098/b361098.txt">Table of n, a(n) for n = 1..16384</a>
%H Michael De Vlieger, <a href="/A361098/a361098.png">Diagram showing k = 1..n for n = 1..54</a> in blue for k counted by A360480(n), in green for k counted by A360543(n), in gold for k counted by A361235(n), and in magenta for k counted by A355432(n). Red dots indicate k | n such that k > 1, while gray dots indicate gcd(k, n) = 1.
%H Michael De Vlieger, <a href="/A361098/a361098_1.png">1016 X 1016 pixel bitmap</a> read left to right in rows, then top to bottom where the k-th pixel is black if A126706(k) is in this sequence, else white (1032256 pixels total).
%e For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p.
%e For prime power n = p^e > 4, e > 0, A360543(n) = p^(e-1) - e, but A360480(n) = A361235(n) = A355432(n) = 0, since the other sequences require omega(n) > 1.
%e For squarefree composite n, A360480(n) >= 1 and A361235(n) >= 1 (the latter for n > 6), but A360543(n) = A355432(n) = 0, since the other sequences require at least 1 prime power factor p^e | n with e > 0.
%e For n = 18, A360480(n) = | {10, 14, 15} | = 3,
%e A360543(n) = | {} | = 0,
%e A361235(n) = | {4, 8, 16} | = 3,
%e A355432(n) = | {12} | = 1.
%e Therefore 18 is not in the sequence.
%e For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10,
%e A360543(n) = | {30} | = 1,
%e A361235(n) = | {8, 16, 27, 32} | = 4,
%e A355432(n) = | {24} | = 1.
%e Therefore 36 is the smallest term in the sequence.
%e Table pertaining to the first 12 terms:
%e Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822;
%e g = A046753 = f + c, tau = A000005, phi = A000010.
%e n | a + b = c | d + e = f | g + tau + phi - 1 = n
%e ------------------------------------------------------
%e 36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 + 9 + 12 - 1 = 36
%e 48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 = 48
%e 50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 + 6 + 20 - 1 = 50
%e 54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 + 8 + 18 - 1 = 54
%e 72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 = 72
%e 75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 + 6 + 40 - 1 = 75
%e 80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 = 80
%e 96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 = 96
%e 98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 + 6 + 42 - 1 = 98
%e 100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 + 9 + 40 - 1 = 100
%e 108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108
%e 112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112
%t nn = 2^16;
%t a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
%t s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &];
%t Reap[ Do[n = s[[j]];
%t If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[[2]]} &@
%t FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]]
%Y Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A355432, A360480, A360543, A361235.
%Y Subsets: A095990\{18}, A286708, A303606, A359280.
%K nonn
%O 1,1
%A _Michael De Vlieger_, Mar 15 2023
|
