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

%I A377311 #10 Oct 31 2024 01:39:08
%S A377311 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,
%T A377311 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,
%U A377311 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
%N A377311 Least positive integer k with k*n primitive practical.
%H A377311 Frank M Jackson, <a href="/A377311/b377311.txt">Table of n, a(n) for n = 1..10000</a>
%e A377311 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.
%t A377311 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]]];
%t A377311 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];
%t A377311 PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
%t A377311 lst={}; Do[m=1; While[!PPracticalQ[n*m],m++]; AppendTo[lst,m],{n,1,100}]; lst
%Y A377311 Cf. A005153, A210445, A267124.
%K A377311 nonn
%O A377311 1,3
%A A377311 _Frank M Jackson_, Oct 24 2024