A317412 Practical numbers (A005153) that are not numbers whose divisors increase by a factor of 2 or less (A174973).

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

Offset: 1

Author

Frank M Jackson, Jul 27 2018

Keywords

• nonn

Comments

Numbers that appear in A005153 but not in A174973.

Examples

			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.
		

Links

• Wikipedia, Practical number.
• 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]

Crossrefs

Cf. A005153, A174973.

Programs

Mathematica

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[#] &]