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A317412
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Practical numbers (A005153) that are not numbers whose divisors increase by a factor of 2 or less (A174973).
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1
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78, 348, 666, 820, 860, 1014, 1272, 1326, 1416, 1464, 1482, 1794, 1830, 2010, 2130, 2190, 2262, 2418, 2628, 2844, 2886, 2988, 3198, 3204, 3320, 3354, 3560, 3666, 3738, 4074, 4134, 4602, 4656, 4758, 4848, 4944, 5136, 5226, 5232, 5424, 5538, 5694, 5886, 6102, 6162, 6328
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OFFSET
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1,1
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COMMENTS
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LINKS
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Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - N. J. A. Sloane, May 20 2023]
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EXAMPLE
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a(1)=78 because it is practical. It has divisors 1, 2, 3, 6, 13, 26, 39, 78, and 13/6 > 2. This is the first occurrence.
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MATHEMATICA
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PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod = prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Dens2DivQ[n_] := Module[{lst=Divisors[n]}, Do[ok=False; If[lst[[m+1]]/lst[[m]]>2, Break[]]; ok=True, {m, 1, Length[lst]-1}]; ok]; Select[Range[10000], PracticalQ[#]&&!Dens2DivQ[#] &]
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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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