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A336507
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Lambda-practical numbers (A336506) that are not phi-practical (A260653).
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2
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45, 135, 225, 405, 675, 765, 855, 1035, 1125, 1215, 1275, 1305, 1395, 1665, 1845, 1935, 2025, 2115, 2295, 2565, 3105, 3375, 3645, 3825, 3915, 4185, 4275, 4995, 5175, 5535, 5625, 5805, 6075, 6345, 6375, 6525, 6885, 6975, 7155, 7695, 7965, 8235, 8325, 9045, 9225
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OFFSET
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1,1
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COMMENTS
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Thompson (2012) proved that all phi-practical numbers are lambda-practical, that all the terms of this sequence are not squarefree numbers, and that this sequence is infinite: for example, 45 * Product_{i=10..k} prime(i) is a term for all k >= 10.
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LINKS
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MATHEMATICA
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phiPracticalQ[n_] := If[n<1, False, If[n==1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok=True; Do[If[lst[[m]]>Sum[lst[[l]], {l, 1, m-1}]+1, (ok=False; Break[])], {m, 1, Length[lst]}]; ok)]]; rep[v_, c_] := Flatten @ Table[ConstantArray[v[[i]], {c[[i]]}], {i, Length[c]}]; lambdaPracticalQ[n_] := Module[{d = Divisors[n], lam, ns, r, x}, lam = CarmichaelLambda[d]; ns = EulerPhi[d]/lam; r = rep[lam, ns]; Min @ Rest @ CoefficientList[Series[Product[1 + x^r[[i]], {i, Length[r]}], {x, 0, n}], x] > 0]; Select[Range[1000], !phiPracticalQ[#] && lambdaPracticalQ[#] &] (* after Frank M Jackson at A260653 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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