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 A361098 Intersection of A360765 and A360768. 8
 36, 48, 50, 54, 72, 75, 80, 96, 98, 100, 108, 112, 135, 144, 147, 160, 162, 189, 192, 196, 200, 216, 224, 225, 240, 242, 245, 250, 252, 270, 288, 294, 300, 320, 324, 336, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). Numbers k such that A360480(k), A360543(k), A361235(k), and A355432(k) are positive. Subset of A126706. All terms are neither prime powers nor squarefree. From Michael De Vlieger, Aug 03 2023: (Start) 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. This sequence contains {A002182 \ A168263}. (End) LINKS Michael De Vlieger, Table of n, a(n) for n = 1..16384 Michael De Vlieger, Diagram showing k = 1..n for n = 1..54 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. Michael De Vlieger, 1016 X 1016 pixel bitmap 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). EXAMPLE For prime p, A360480(p) = A360543(p) = A361235(p) = A355432(p) = 0, since k < p is coprime to p. 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. 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. For n = 18, A360480(n) = | {10, 14, 15} | = 3, A360543(n) = | {} | = 0, A361235(n) = | {4, 8, 16} | = 3, A355432(n) = | {12} | = 1. Therefore 18 is not in the sequence. For n = 36, A360480(n) = | {10, 14, 15, 20, 21, 22, 26, 28, 33, 34} | = 10, A360543(n) = | {30} | = 1, A361235(n) = | {8, 16, 27, 32} | = 4, A355432(n) = | {24} | = 1. Therefore 36 is the smallest term in the sequence. Table pertaining to the first 12 terms: Key: a = A360480, b = A360543, c = A243823; d = A361235, e = A355432, f = A243822; g = A046753 = f + c, tau = A000005, phi = A000010. n | a + b = c | d + e = f | g + tau + phi - 1 = n ------------------------------------------------------ 36 | 10 + 1 = 11 | 4 + 1 = 5 | 16 + 9 + 12 - 1 = 36 48 | 16 + 2 = 18 | 3 + 2 = 5 | 23 + 10 + 16 - 1 = 48 50 | 18 + 1 = 19 | 4 + 2 = 6 | 25 + 6 + 20 - 1 = 50 54 | 19 + 2 = 21 | 4 + 4 = 8 | 29 + 8 + 18 - 1 = 54 72 | 27 + 4 = 31 | 4 + 2 = 6 | 37 + 12 + 24 - 1 = 72 75 | 25 + 2 = 27 | 2 + 1 = 3 | 30 + 6 + 40 - 1 = 75 80 | 32 + 3 = 35 | 3 + 1 = 4 | 39 + 10 + 32 - 1 = 80 96 | 38 + 7 = 45 | 4 + 4 = 8 | 53 + 12 + 32 - 1 = 96 98 | 41 + 3 = 44 | 5 + 2 = 7 | 51 + 6 + 42 - 1 = 98 100 | 42 + 4 = 46 | 4 + 2 = 6 | 52 + 9 + 40 - 1 = 100 108 | 44 + 8 = 52 | 5 + 4 = 9 | 61 + 12 + 36 - 1 = 108 112 | 48 + 3 = 51 | 3 + 1 = 4 | 55 + 10 + 48 - 1 = 112 MATHEMATICA nn = 2^16; a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p]; s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]; Reap[ Do[n = s[[j]]; If[And[#1*a053669[n] < n, n/#1 >= #2] & @@ {Times @@ #, #[]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}]][[-1, -1]] CROSSREFS Cf. A000005, A000010, A001694, A002182, A007947, A045763, A053669, A119288, A126706, A168263, A243822, A243823, A286708, A303606, A355432, A359280, A360480, A360543, A361235. Sequence in context: A083674 A160063 A260927 * A291713 A287840 A244326 Adjacent sequences: A361095 A361096 A361097 * A361099 A361100 A361101 KEYWORD nonn AUTHOR Michael De Vlieger, Mar 15 2023 STATUS approved

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Last modified December 9 04:08 EST 2023. Contains 367681 sequences. (Running on oeis4.)