A258804 a(n) = gpf(a(n-3))*gpf(a(n-2)) + gpf(a(n-1)), with a(1)=a(2)=1 and a(3)=2 and where gpf(n) is the greatest prime dividing n, A006530(n).

1, 1, 2, 3, 5, 11, 26, 68, 160, 226, 198, 576, 1246, 122, 328, 5470, 3048, 22554, 69648, 24184, 262752, 4386396, 70190, 22222, 4639830, 1914046, 3227106, 35917950, 2738592, 325870, 124850, 8375086, 7397758, 129192, 6948110, 10496178, 82166, 740450, 44446188, 1473852, 184450, 342342, 14496, 740, 2906, 7040, 53772, 20464

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

Cf. A006530.

Sequence in context: A358554 A369495 A273755 * A006888 A009589 A098179

Adjacent sequences: A258801 A258802 A258803 * A258805 A258806 A258807

Keyword

nonn,changed

Author

Iago Casabiell González, Sep 22 2015

Status

approved