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 A364702 Numbers k in A361098 that are not divisible by A007947(k)^2. 5
 48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, 363, 375, 378, 384, 396, 405, 416, 448, 450, 468, 480, 486, 490, 504, 507, 525, 528, 540, 550, 560, 567, 578, 588, 594, 600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1. Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098. Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T. This sequence gives k that are in S but not in T. LINKS Michael De Vlieger, Table of n, a(n) for n = 1..10000 FORMULA This sequence is A361098 \ A286708. EXAMPLE Let B = A126706. B(1) = 12 is not in the sequence since 3*6 > 12. B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18. B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1). B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48. B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc. MATHEMATICA nn = 2^10; 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, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]] CROSSREFS Cf. A001221, A001222, A001694, A007947, A053669, A119288, A126706, A286708, A361098. Sequence in context: A327894 A332242 A296872 * A357429 A258694 A328738 Adjacent sequences: A364699 A364700 A364701 * A364703 A364704 A364705 KEYWORD nonn AUTHOR Michael De Vlieger, Aug 03 2023 STATUS approved

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Last modified May 24 05:24 EDT 2024. Contains 372772 sequences. (Running on oeis4.)