cp's OEIS Frontend

A361128 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).

%I A361128 #34 Mar 10 2023 07:48:52
%S A361128 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 A361128 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 A361128 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 A361128 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 A361128 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 A361128 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 A361128 By definition of A360519, the set of primes in L(n) is disjoint from the primes in R(n).
%H A361128 N. J. A. Sloane, <a href="/A361128/b361128.txt">Table of n, a(n) for n = 2..20000</a>
%p A361128 # Suppose bW is a list of the terms of A360519.
%p A361128 # Then f3(bW[n-1], bW[n], bW[n+1]); returns [L(n), M(n), R(n)] where:
%p A361128 with(numtheory);
%p A361128 f3:=proc(a,b,c)
%p A361128 local lefta,midb,rightc,i,p,pa,pc,ta,tb,tc,t1,t2;
%p A361128 ta:=a; tb:=b; tc:=c;
%p A361128 # left
%p A361128 t1:=igcd(a,b);
%p A361128 t2:=factorset(t1);
%p A361128 t2:=convert(t2,list);
%p A361128 lefta:=1;
%p A361128 for i from 1 to nops(t2) do
%p A361128 p:=t2[i];
%p A361128 while (tb mod p) = 0 do lefta:=lefta*p; tb:=tb/p; od;
%p A361128 od:
%p A361128 # right
%p A361128 t1:=igcd(b,c);
%p A361128 t2:=factorset(t1);
%p A361128 t2:=convert(t2,list);
%p A361128 rightc:=1;
%p A361128 for i from 1 to nops(t2) do
%p A361128 p:=t2[i];
%p A361128 while (tb mod p) = 0 do rightc:=rightc*p; tb:=tb/p; od;
%p A361128 od:
%p A361128 # middle
%p A361128 midb:=b/(lefta*rightc);
%p A361128 [lefta,midb,rightc];
%p A361128 end; # _N. J. A. Sloane_, Mar 09 2023
%Y A361128 Cf. A360519, A361118, A361129, A361130.
%K A361128 nonn
%O A361128 2,2
%A A361128 _Scott R. Shannon_, _Rémy Sigrist_, and _N. J. A. Sloane_, Mar 09 2023