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

1,3

Links

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

Example

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.

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

Crossrefs

Cf. A005153, A210445, A267124.

Sequence in context: A373427 A254881 A100946 * A200019 A282824 A106664

Adjacent sequences: A377308 A377309 A377310 * A377312 A377313 A377314

Keyword

nonn

Author

Frank M Jackson, Oct 24 2024

Status

approved