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).

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

Offset: 2

Author

Scott R. Shannon, Rémy Sigrist, and N. J. A. Sloane, Mar 09 2023

Keywords

• nonn

Comments

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.

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).

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

Links

• N. J. A. Sloane, Table of n, a(n) for n = 2..20000

Crossrefs

Cf. A360519, A361118, A361129, A361130.

Programs

Maple

# Suppose bW is a list of the terms of A360519.
# Then f3(bW[n-1], bW[n], bW[n+1]); returns [L(n), M(n), R(n)] where:
with(numtheory);
f3:=proc(a,b,c)
local lefta,midb,rightc,i,p,pa,pc,ta,tb,tc,t1,t2;
ta:=a; tb:=b; tc:=c;
# left
t1:=igcd(a,b);
t2:=factorset(t1);
t2:=convert(t2,list);
lefta:=1;
for i from 1 to nops(t2) do
p:=t2[i];
while (tb mod p) = 0 do lefta:=lefta*p; tb:=tb/p; od;
od:
# right
t1:=igcd(b,c);
t2:=factorset(t1);
t2:=convert(t2,list);
rightc:=1;
for i from 1 to nops(t2) do
p:=t2[i];
while (tb mod p) = 0 do rightc:=rightc*p; tb:=tb/p; od;
od:
# middle
midb:=b/(lefta*rightc);
[lefta,midb,rightc];
end; # N. J. A. Sloane, Mar 09 2023