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A379325 Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.

%I A379325 #24 Feb 09 2025 20:05:27
%S A379325 1,2,2,6,2,6,2,6,20,6,28,30,2,6,20,42,6,6,28,30,2,66,6,78,20,42,88,30,
%T A379325 6,20,104,6,28,30,42,2,66,140,6,30,78,20,6,42,88,30,6,28,66,20,204,
%U A379325 104,210,6,220,28,228,78,30,42,2,260,66,30,272,276,140,6,42,30,304,306,308,78,20,6,330,42,340,342,348,88,30,364,368,42,380,6,390,28,66
%N A379325 Largest primitive practical number p that divides the n-th practical number - A005153(n) such that the radical of the quotient A005153(n)/p is a divisor of p.
%C A379325 The relationship between a practical number and the largest primitive practical number that divides it such that the radical of the quotient is a divisor of same primitive provides an equivalence relation. Practical numbers that have the same progenitive primitive. This property arises from the characteristic of practical numbers and, in particular, primitives that says a practical number multiplied by power combinations of any of its divisors is also practical. This sequence identifies the primitive progenitor of each practical number A005153(n).
%C A379325 Note that this sequence and A378202 are similar but the first difference is at a(63) as explained in the example.
%H A379325 Frank M Jackson, <a href="/A379325/b379325.txt">Table of n, a(n) for n = 1..10000</a>
%e A379325 a(63) = 66. A005153(63) = 264 and the largest primitive practical number that divides the practical number 264 is 88. However the radical of the quotient 264/88 is 3 and 3 is not a divisor of 88. The next greatest primitive divisor of 264 is 66 and the radical of the quotient 264/66 is 2 and 2 is a divisor of 66.
%e A379325 a(131) = 306. A005153(131) = 612 and it is divisible by two primitive practical numbers 204 and 306 with their quotient a divisor of their primitive in both cases but 306 is chosen as the larger primitive.
%t A379325 plst=Last/@ReadList["https://oeis.org/A005153/b005153.txt", {Number, Number}]; pplst=Last/@ReadList["https://oeis.org/A267124/b267124.txt", {Number, Number}]; Rad[n_] := Times @@ First /@ FactorInteger[n]; getpplst[n_] := Module[{}, Select[pplst, #<=n &]]; lst1={}; Do[lst=getpplst[plst[[n]]]; lnh=Length@lst; m=0; While[Mod[j=plst[[n]], k=lst[[lnh-m]]]!=0||Mod[k, Rad[j/k]]!=0, m++]; AppendTo[lst1, {j, k}], {n, 1, 100}]; Last/@lst1
%Y A379325 Cf. A005153, A007947, A267124, A377377.
%K A379325 nonn
%O A379325 1,2
%A A379325 _Frank M Jackson_, Dec 20 2024