A377311 Least positive integer k with k*n primitive practical.

1, 1, 2, 5, 4, 1, 4, 11, 34, 2, 6, 17, 6, 2, 2, 17, 12, 17, 12, 1, 2, 3, 12, 31, 188, 3, 82, 1, 12, 1, 16, 37, 2, 6, 4, 41, 18, 6, 2, 47, 20, 1, 20, 2, 158, 6, 24, 67, 236, 94, 4, 2, 24, 41, 4, 59, 4, 6, 24, 79, 24, 8, 202, 67, 4, 1, 30, 3, 4, 2, 30, 97, 30, 9, 158, 3, 4, 1, 36, 97, 254, 10, 36, 101, 4, 10, 4, 1, 36, 79, 4, 3, 6, 12, 4, 127, 42, 118, 298, 47

Offset: 1

Author

Frank M Jackson, Oct 24 2024

Keywords

• nonn

Examples

			a(9) = 34. Consider the following sequence of 16 even multiples of 9 namely (18, 36, 54, . . . , 288, 306), all are practical numbers but only 9*34 = 306 is a primitive practical number. This is because 306 when divided by 3 is no longer practical whereas the other 15 even multiples remain practical when divided by 3.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005153, A210445, A267124.

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]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n],#[[2]]>1&],m,ok=False},Do[If[!PracticalQ[n/plst[[m]]],ok=True,ok=False; Break[]],{m,1,Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
lst={}; Do[m=1; While[!PPracticalQ[n*m],m++]; AppendTo[lst,m],{n,1,100}]; lst