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

1,1

Comments

Numbers that appear in A005153 but not in A174973.

Links

Table of n, a(n) for n=1..46.

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]

Example

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.

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[#] &]

Crossrefs

Cf. A005153, A174973.

Sequence in context: A118938 A362546 A206004 * A231393 A231461 A007255

Adjacent sequences: A317409 A317410 A317411 * A317413 A317414 A317415

Keyword

nonn

Author

Frank M Jackson, Jul 27 2018

Status

approved