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Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives L(n).
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%I #34 Mar 10 2023 07:48:52

%S 1,2,5,7,3,4,5,11,9,2,7,11,3,5,2,11,13,3,4,7,13,5,2,17,7,9,2,13,17,3,

%T 2,19,5,3,4,11,17,25,2,23,3,19,4,13,3,5,2,29,3,31,8,7,3,37,4,17,3,41,

%U 16,5,23,7,12,5,29,49,2,3,43,25,2,3,47,5,8,3,7,19,2,27,5,31

%N Let b = A360519; let Lg = gcd(b(n-1),b(n)), Rg = gcd(b(n),b(n+1)); let L(n) = prod_{primes p|Lg} p-part of b(n), R(n) = prod_{primes p|Rg} p-part of b(n), M(n) = b(n)/(L(n)*R(n)); sequence gives L(n).

%C The p-part of a number k is the highest power of p that divides k. For example, the 2-part of 24 is 8, the 3-part is 3.

%C One can think of A360519 as a chain of circles, each circle linked to its neighbors to the left and the right. The n-th term b(n) = A360519(n) is a product L(n)*M(n)*R(n), where L(n) is the part of b(n) sharing primes with the term to the left, R(n) the part sharing primes with the term to the right, and M(n) is the rest of b(n).

%C By definition of A360519, the set of primes in L(n) is disjoint from the primes in R(n).

%H N. J. A. Sloane, <a href="/A361128/b361128.txt">Table of n, a(n) for n = 2..20000</a>

%p # Suppose bW is a list of the terms of A360519.

%p # Then f3(bW[n-1], bW[n], bW[n+1]); returns [L(n), M(n), R(n)] where:

%p with(numtheory);

%p f3:=proc(a,b,c)

%p local lefta,midb,rightc,i,p,pa,pc,ta,tb,tc,t1,t2;

%p ta:=a; tb:=b; tc:=c;

%p # left

%p t1:=igcd(a,b);

%p t2:=factorset(t1);

%p t2:=convert(t2,list);

%p lefta:=1;

%p for i from 1 to nops(t2) do

%p p:=t2[i];

%p while (tb mod p) = 0 do lefta:=lefta*p; tb:=tb/p; od;

%p od:

%p # right

%p t1:=igcd(b,c);

%p t2:=factorset(t1);

%p t2:=convert(t2,list);

%p rightc:=1;

%p for i from 1 to nops(t2) do

%p p:=t2[i];

%p while (tb mod p) = 0 do rightc:=rightc*p; tb:=tb/p; od;

%p od:

%p # middle

%p midb:=b/(lefta*rightc);

%p [lefta,midb,rightc];

%p end; # _N. J. A. Sloane_, Mar 09 2023

%Y Cf. A360519, A361118, A361129, A361130.

%K nonn

%O 2,2

%A _Scott R. Shannon_, _Rémy Sigrist_, and _N. J. A. Sloane_, Mar 09 2023