OFFSET
1,3
COMMENTS
The sequence appears to oscillate about its mean, with greater amplitude each time (an empirical observation).
The sequence seems to always give a value for a(n) that is relatively prime to each of g(n-1), g(n-2), and g(n-3), where g(i) = gpf(a(i)) (empirical observation).
Proof of the second observation: If a(n) is relatively prime to g(n-1), g(n-2), and g(n-3), then so is g(n). In particular, if g(n) is relatively prime to g(n-1) and g(n-2), then g(n-2)*g(n-1) + g(n) = a(n+1) is relatively prime to g(i) for i=n-2,n-1,n. - Danny Rorabaugh, Dec 01 2015
LINKS
Sean A. Irvine, Table of n, a(n) for n = 1..87 (terms 1..80 from Robert Israel)
EXAMPLE
Let gpf(x) = A006530(x).
a(4) = gpf(a(1))*gpf(a(2)) + gpf(a(3)) = 1*1 + 2 = 3.
a(5) = gpf(a(2))*gpf(a(3)) + gpf(a(4)) = 1*2 + 3 = 5.
MAPLE
gpf:= x -> max(numtheory:-factorset(x)):
gpf(1):= 1:
a:= proc(n) option remember;
gpf(procname(n-3))*gpf(procname(n-2))+gpf(procname(n-1))
end proc:
a(1):= 1: a(2):= 1: a(3):=2:
seq(a(n), n=1..70); # Robert Israel, Dec 01 2015
MATHEMATICA
gpf[n_] := FactorInteger[n][[-1, 1]];
a[n_] := a[n] = Switch[n, 1, 1, 2, 1, 3, 2, _, gpf[a[n-3]] * gpf[a[n-2]] + gpf[a[n-1]]];
Array[a, 70] (* Jean-François Alcover, Aug 26 2022 *)
PROG
(PARI) gpf(n) = if (n==1, 1, vecmax(factor(n)[, 1]));
lista(nn) = {print1(b = 1, ", "); print1(c = 1, ", "); print1(d = 2, ", "); for (n=1, nn, e = gpf(b)*gpf(c) + gpf(d); print1(e, ", "); b = c; c = d; d = e; ); } \\ Michel Marcus, Oct 07 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Iago Casabiell González, Sep 22 2015
STATUS
approved